How big? What do you want to grow? Indoors or outdoors? There are so many variables available to you that you need to narrow down your requirements to get an answer that will help you.
A media system consists of fish tank, grow bed(s), possibly a sump, pump, siphon(s) or indexing valve, various plumbing, media to fill beds (many choices), fish, fish food, plants. If indoors, there is lighting to consider. If outdoors, will you need a greenhouse? You'll need to think about a design which fits your situation and goals and then start about thinking components and pricing those components. Systems run from relatively cheap to outrageously expensive. There are also raft systems, NFT, and towers. You may need to study the subject for a while. Compared to dirt gardening, this requires more of an investment. You can build a system yourself or buy an off-the-shelf system.
There is a book entitled "Aquaponic Gardening" by Sylvia Berstein, which contains a lot of information in one place or you can study this site and several others to get information. There are videos on the subject too.
Do you want to build it yourself for buy a pre-made system? if you want to buy the owner of this site has systems for sale that will work for your requriements: http://www.theaquaponicstore.com/AquaBundance-Modular-Sleek-Aquapon...
I recommend spending a fair amount of time on youtube where you will see many different designs, you can start on my channel to see how I built my system and go from there. www.youtube.com/aquaponicsnyc. You might find barrelponics to be a solution that is easy to execute which you will find on youtube as well.
I am new, and interested in this subject as well. I think the answer to cost is that you can spend as much or as little as you would like, proportional to how much work you are willing to put in it. It's relatively easy to buy a system that is ready to go, but it will cost a pretty penny. On the other hand, you might build a system for minimal investment if you use sites like Freecycle and Craigslist, and are willing to really scout for supplies. Myself, I am looking to grow tomatoes, cucumbers, some lettuces, and a few other things (I would love to do cantaloupe and watermelon, but don't see how that could be easy to do in an aquaponics system), as well as tilapia, and maybe some bass. (Yes, the bass will eat some of the tilapia, I know--I just want a few bass, and a lot of tilapia, and I won't put them in the same place until the tilapia are big enough? smart enough? fast enough? to get away/hide.) I am also in the process of building a greenhouse outdoors, so that will most likely be where my system is located. I would love to set up in an old bathtub or food-grade barrel, if I can find one. One thing I am curious about is fish density. I have read conflicting things. One article said 7 fish per gallon of water, and another said 55 lbs of fish per 250 gallons, which is .22 lbs per gallon. I am pretty confused, as that seems to be a pretty big difference in numbers of fish. I guess, though, if you had 7 fingerlings, they could weigh around .22 lbs, maybe? I am looking to start small, like maybe 50-100 gallons. Anyone that knows anything about this, any info you can provide would be most appreciated! Anyhow, Hailey, good luck with your system!