I opened up my first bank account when I was about eight. Christmas and birthday presents sometimes came in checks or cash, so they got deposited into my savings account. I loved watching my money grow, and have been a saver ever since. There is a great satisfaction in watching your money multiply with compounded interest, while all you do is leave it alone and give it time.
My choice for my savings account now is ING Direct. They have CD's and checking accounts for individuals and business which offer better interest rates for lower minimums than any brick and mortar bank. Currently their savings account rate is 3% with no minimum and it's insured by the FDIC! If you have any amount of money sitting in your checking account that you are not using, you owe it to yourself to move it to this savings account and allow compound interest to make you some money.
You can set up an automatic savings plan that will make your money grow without you having to remember to put it aside every month. Even if you move only $100 (that's about 30 days of buying one vente mocha a day), the compounding interest will eventually grow enough to cover your checking account fees at your other bank, and you will have $1200 (not counting interest) at the end of the year.
Or better yet, move your checking to an ING checking account. Their checking accounts still earn higher interest with lower minimums than you are surely getting at your bank now (unless you have several million dollars at your bank, in which case, you need to move $100,000 over to ING Direct to keep it FDIC insured), and spread the rest to other FDIC insured banking institutions to keep yourself insured under the FDIC limit of $100,000 per account holder.
If you don't feel comfortable with an online bank (they do have internet cafe locations in various states where you can go in and get help setting up or maintaining your account), you should take into consideration that ING is a big and well-known Dutch bank which is only limited by US banking regulations from proliferating to every block to keep down competition to local US banks. Being a Dutch bank has advantages because the Dutch are very conservative; they would always be very strict with regulations concerning money (no sub-prime loans would ever be made, much less sold in Holland). Countries whose cultures encourage saving, will protect the savings.
If you still don't want to open an online account, then walk into your nearest credit union (nearly anyone can open an account in a credit union, some require only that you are a resident of the city or state where you live) and open a savings account. Unlike a commercial bank, you become part of a credit union, so your money is used for loans to other members and their money can be used as a loan to you. In a credit union, your money will be working locally and you are setting up a relationship that could help you with your next home mortgage or auto loan.
Saving your money in a safe place is important, but just as important is doing it in a smart place where it can earn you a good return and / or future benefits.