Hi:
The amount of bandwidth TeamTalk uses, especially on large servers where people use voice activation, is absolutely insanely huge. You will need a VPS that allows for at least 500 GB of network traffic per month. You're not going to get this for free. Also, all of the free hosts do not generally allow servers. However, memory and disc space for TeamTalk are not a huge issue. The best you're going to do is check:
http://www.lowendbox.com/
every once in a while. You should be able to get a deal on a server that will fit your needs for between $2 and $5 US per month. You should also expect to spend another $4 per year for a domain. I don't know what your situation is, but you should be able to earn this amount of money by doing odd jobs for people, or perhaps as an allowance from your parents. I know as a pre-teen, I had chores around the house and got $3 a week for completing them.
Note that the prices above assume you want to run team talk, *only*! If you want to run e-mail servers, and perhaps a website with PHP and a database, you're looking at a price of between $10 and $20 a month, as you need a lot more memory. Your best option might be to pay for a TeamTalk server, run it as teamtalk.YourDomain.com, and then find a free web host to point www.YourDomain.com at. Some free web hosts will allow you to have PHP and perhaps 1 database. I've heard good things about:
http://www.000webhost.com/
but I don't know if you can point your own domain at them or not.
If you want to do email on your domain, now that Google Apps is no longer free, your cheapest option will run you about $40 a year:
https://www.fastmail.fm/signup/personal.html
I have used these guys in the past, and they do what they do well.
I know the frustration of being a teenager and wishing everything was free. Unfortunately, it just doesn't work that way. I honestly do believe that $100 per year for TeamTalk, e-mail, and web hosting, is the best you can possibly get. You're also going to learn by doing valuable things about Linux administration, DNS, etc, at a much cheaper price than any course you could take. Most web hosts and email providers will be happy to take yearly payments by certified cheque or money order, so you should be able to get away without a credit card.
All of the above kind of assumes you're a middle class North American. Perhaps your resources are somewhat less than I have assumed above. If so, your best bet is going to be to find a partner who wants to run a website and server with you, and can contribute the money, while you put in the time.