If you are at least 18, you should be able to open an account in your own name alone. Pretty much all you need to do is go to the bank with some money and tell them you want to open an account and what types of accounts. You'll have to fill out some forms (or they might just type the information straight into their computer) and provide social security number and some photo ID to prove you are really you.
Personally I prefer credit unions to banks because many banks charge monthly fees, etc., particularly on low balance accounts while credit unions generally charge fewer fees and pay slightly better interest rates on the savings and checking accounts.
You should probably check on the amount of fees before you open an account because if you're only putting a small amount of money in there, the bank fees could eat up a large chunk of it. And there's probably a fee for closing the account so you don't want to wait until you've already opened one to find out it costs too much to be worth it.
Be aware that unless you are planning to file your own tax return without help from your parents, they will find out you have this account because you will have to include any interest you get from it on your tax return. Of course, with the trivial interest rates being paid these days, the interest will likely only be a small amount, so if you don't have a job or enough other income to be required to file a return, that won't be a problem.