All plants need a grow medium - a physical substance the plant can grow roots into, stabilize itself, and pull nutrients from.  Here are some options:

Soil is the most common and often least scary as we're all pretty used to the fact that plants grow in the ground.  There's lots of fancy soils you can buy at local plant nursery's and many are tailored for specific types of plants and promote their growth or flowering or fruiting or you-name-it.  The pro's of soil are how common it is, that it's prime for root vegetables, and that plants love it.  The con's are that bugs love it and it is dirty - no joke intended.  If you buy a bag of soil today, let it sit in negative temperatures over winter, and then plant that soil in a sealed clean-room down at NASA at the start of Spring, within a week it will have bugs in it.  Maybe a profound entomologist can explain it better but to us it's magic.  Soil = bugs. This is bad news for small, enclosed, greenhouses because they are a paradise for plant-eating insects as all their natural predators are kept out.  Infestations can come in the blink of an eye and explode to unfathomable populations due to this.  Because of these factors, we recommend soil for root vegetables and in a separate greenhouse, or outside, so that more active forms of pest-control can be utilized like the releasing of Ladybugs.  In closing, soil is great if you're willing to deal with bugs and dirt.  If you'd like to have your plants inside and pick-n-eat ready with very little cleaning, then soil isn't for you.

Clay Pebbles are an excellent grow medium for hydroponics.  They give the plant the physical aspects of soil: something to grab hold of and build root structures around.  They also do a great job of staying sanitary and bug-free.  The pro's of Clay Pebbles are how much plants like it and how little bugs like it.  The con's are that it's more difficult to start seedlings in clay pebbles as they less-than-accurately recreate the conditions of soil.  Clay pebbles don't hold the moisture against the seed like other grow mediums.  Also, they don't immediately give it a stable resting place so some seeds and when germinating, they will push themselves off the dome of a clay pebble into the nutrient solution below.  Therefore, we view clay pebbles as an excellent medium for seedlings or maturing plants.

Rockwool is supreme for the hydroponic germination process.  It allows just a little light through while holding nutrient water close to the seed.  It also drains and allows oxygen in, enabling strong and rapid plant growth.  It most closely resembles the physical structure of soil while simultaneously keeping the anti-insect properties of Clay Pebbles.  Also, seeds cleanly stay imbedded inside it providing a prefect habitat for growth.  We've found the mats with pre-sectioned 1.5 inch cubes to be the most effective, especially if it has pre-drilled seed holes.  There's other types like little cubes that are closer to the size of a cubic centimeter that we've tried and were dissatisfied with.  At that dimension they really just try to act like Clay Pebbles but they start falling apart and leave little for the plant to grab onto.  When that happens, as the plant grows, its top gets too heavy for the structure below and it just falls over inside your net pot.  The pro's are the ability for seeds to germinate within it and how bugs do not like to live within it.  The con's are how easily algae can grow on it, smother the germinating seed, and result in a place for a select few insects to nest.  This is especially true if left exposed to light.  Because of this, we find Rockwool to be excellent for the germination phase specifically.

A Mixture is our favorite and how we'll grow for the foreseeable future.  We've learned to use the Rockwool as an initial germination medium and then later transplant the Rockwool cube with seedling intact, into clay pebbles.  We buy 1020 tray sized Rockwool sheets and plant rows/columns of whatever we like.  We then use a butterknife to cut the Rockwool sheet as needed to remove larger seedlings that have sprouted and transplant them into 3" net pots.  We put the block of Rockwool into the base of the net-pot, fill the rest of the net pot with Clay Pebbles, and then fill the wide-mouth mason jar with nutrient-rich water.  The plant has plenty of medium, in both Rockwool and Clay Pebbles, to grow its roots through and physically stabilize itself for maturation.  The Clay Pebbles also do a good job of holding the Rockwool stationary inside the net pot for when you need to relocate for organizing or harvesting.  They also shield the Rockwool from the light and therefor prevent algae growth.  The Rockwool stays damp and prime for germination, bathing the seed in nutrient rich solution without drowning it.  The only con is that it takes some patience and a bit of planning to do it right.  None the less, it's the best of all worlds and by far our most recommended.

The Aztecs used forms of hydroponics in the 10th century with gardens that floated on lakes, the Chinese did it in the 13th century for rice fields in bad soil, and the earliest known hydroponics based scientific research was recorded in the 16th century.   Hydroponics have been around a lot longer than most imagine and the processes and techniques have massively advanced in the last few decades.  Similarly, thanks mainly to the marijuana industry, the technology for hydroponics has become smaller, cheaper, and more available to average consumers.  With todays systems, hydroponic growers have reported up to 50% faster growth during the vegetative stage with up to 90% less water consumption.  The numbers are incredible and the systems are definitely proven.  Now, those growers are massive industrial farmers who've spent probably millions on their hydroponic systems and push the forefront of hydroponic technologies to the next level.  We're not quite on that level, we don't want to spend that much, and we're just looking for a good way to supplement our family's food intake with some clean, fresh, homegrown fruits and vegetables.  So, we're not going to see those numbers, exactly, but we can sure get close!  Personally, we've noticed about 70% more efficient water consumption and there's no doubt the growth is more rapid.  Also, growing hydroponically eliminates most bug issues by simply removing the soil that they often nest and breed in.  Plants grow more consistently in hydroponics and quite honestly taste better.  We grow plants to maturity in hydroponics and we also start and acclimate seedlings for transplant hydroponically.  We fully recommend trying it at a small scale just to experience the difference.  

There are a few new things to think about when you're looking to grow hydroponically:

Nutrient Solutions: Soil usually has some nutrients in it already but most of us still top if off with things like fertilizer or Miracle-Gro.  Similarly, for any sort of hydroponics, nutrients need to be added to the water to create what's called a Nutrient Solution. There are many different brands and types of solutions out there but the baseline idea remains the same: feed the plants what they need so they can grow. There's really nothing advanced to it and it's a very clean process.  The concentrated nutrient powder we most often use, Max-i-gro, comes with a little scooper and the directions say 1 scoop per gallon.  We simply dump a scoop in our little garden sprayer and use it to refill our grow jars.  That's all.  Nice and easy.  A tip would be to remove the little nozzle at the end of the sprayer so the solution comes out as a stream.

Algae: The closer the water is to sunlight, the more algae will grow.  In really bad circumstances, algae can suffocate smaller plants by starving their roots of nutrients and oxygen. Similarly, bugs like fruit flies will lay eggs in it so we advised you to be careful when watering to not promote much algae.  The key solution is to keep the water level under the light line.  If the water isn't touched by the light, the algae as much less opportunity to grow. This is why we cover our jars so that light doesn't hit the solution within or any roots.  Depending on the grow jar, we use koozies, black socks, paint, etc to shield the jar's interior. It's wise to keep an eye on algae buildup throughout any grow and if you see algae building up, immediately scrape it off or remove 'infected' parts and cover the rest with clean clay pebbles.

Cleaning: Algae, minerals, plant debris, and other stuff will inevitably build up on your growing medium and in your grow jars or whatever hydroponics system you're running. We like to keep things pretty clean so we swap out our grow jars here and there and run them through the dishwasher.  If we have particularly bad buildup, we'll soak it with some citric acid for a day or two, give it a scrub and a rinse, and back through the dishwasher.  You'll notice more buildup than usual if your water line is regularly too high and/or if you have hard water.

Plants need nutrients to grow and water to allow those nutrients to be absorbed and distributed.  In nature this is normally accomplished by rain soaking into nutrient filled soils that plants are growing in.  Growing with soil in your greenhouse is similar but you provide the rain and you replenish the nutrients with new soils or plant food supplements.  Growing in Clay Pebbles or Rockwool is a bit different as they don't hold any nutrients but simply act as a structural medium for plants to take hold of.  This is where Nutrient Solutions come into play.  Nutrient Solutions are often made of concentrated minerals like calcium, nitrogen, and phosphorus as well as some chemicals for balancing pH levels.  Some are tailored for growth and others for bloom but generally speaking, they are all pretty much all the same thing.  You can get them dry like a powder or in concentrated liquid form.  Hydroponics are the methods of distributing these nutrient solutions to your plants and they come in active and passive varieties.  Active systems have pumps, filters, and automated equipment while passive systems are usually just manually refilled.  We're not going to go over all the different types of hydroponics systems as there's plenty of websites out there dedicated to that information.  Here we'll just give an overview of our experiences and what we currently use.

Nutrient Film Technique (NFT) is an active method where nutrient solution flows like a river to plants in a line via a pump that pulls from a reservoir.  Most easily made from PVC pipe with holes cut in it for net pots to be placed at different distances from each other depending on plant type and space requirements.  A con is that if these systems aren't flushed consistently, once every few months, you can get buildup of things like algae and bacteria.  Flushing is removing all the plants, running a diluted bleach solution through for a little while, wiping down everything, and repeating those two steps until it's clean.  Then, you dump the bleach solution out and rinse the reservoir, pump, and pipes by filling your reservoirs with clean water, running it for about as long as you ran the bleach solution, and then dumping that, and then doing it again.  Finally, you refill the reservoir with nutrient solution and put your plants back in.  It seems like a lot of work but it's really not that bad a couple times a year.  The pump itself does great here at emptying the reservoir and giving you a way to spray down parts of the system.  The biggest issue is that you need a place for the plants to go while flushing the system.  Don't worry, we've got a stand for that!  We recommend this system if you're growing many of the same plant as harvest times will be more synchronized across entire runs.  We remove the plants when we harvest which is half of the work needed to flush the system.  Usually, this technique has a higher level of automation making it great for growers who want to spend less time throughout the week watering.  This system does best with 4'x8' footprints or greater.

Ebb and Flow is an active method where a pump pulls nutrient solution from a large main reservoir and fills a number of smaller reservoirs, usually something like 5 gallon buckets, for a set period of time.  Then, the pump reverses and returns the solution to the main reservoir.  This system doesn't necessarily need aeration as the plants get oxygen from when the pump is reversed and the roots are exposed to air.  Usually this system is used for larger plants, even trees.  Flushing this system is a similar process to the NFT system except it has less pipes and more buckets.  As far as placing the plants somewhere when flushing, we recommend just having extra buckets and doing bucket rotations every so often.  Similar to the NFT systems, this is fairly automated and is great for growers with busy schedules.  We recommend this system if you want to grow big or specialized plants.  This system does best with 8'x8' footprints or greater.

Deep Water Culture can be an active or passive method of hydroponics where the plant's roots are suspended in a nutrient solution.  It's a reservoir with a lid that has holes for net pots to sit in and you fill the reservoir to about the middle of the net pots.  Similar to the Nutrient Film Technique, the plants benefit from an aerated solution.  It's a super simple system and can be very inexpensive.  Active methods have floats and pumps that keep the water at the desired level but this will dilute the reservoir if you don't refill the nutrients.  It could also increase your need for aeration as the roots will not get the Ebb and Flow effect as solution is consumed and the level drops exposing the roots to open air.  Some systems have special floats that the net pots sit in, keeping them always at the right water level.  These systems can range in size from things like large plastic storage bins down to cups.  The larger systems often require less frequent manual watering.  The Kratky Method is a variation that uses wide mouth mason jars, which is what we've based our grow method on.   We love this system because it make harvesting and rearranging plants super easy.  The ability to simply rotate, reposition, or relocate plants is a major bonus for our preferred gardening style.  The jars are easy to clean on a per-plant basis and having root systems separate helps prevent the spread of disease.  Also, mason jars are perfect size for the shelving in our smaller indoor units and just right for lots of fruits and vegetables like different varieties of lettuce, berries, peppers, herbs, and much more!  Watering is a little more hands on as mason jars only come so large.  For 16oz jars, we were refilling different lettuces every 4-6 days, chives 7-10 days, peppers 5-8 days, and raspberries about every 2 days.  We've switched some of those to 32oz jars and, as you'd expect, pretty much doubled those times.  This is by far our most recommended method and we have starter kits and guides to help you set up and get growing!  This system does best at any footprint.