Online banking has many advantages and few dis-advantages.
Online Banking Myths
Before you change banks, take a look at how you use your bank. Some of the disadvantages are that you have to mail your check to the bank, you don't have a site where you can talk to someone face-to-face and you have to make sure you can use ATMs without paying a fee. On the plus side, you have someone available by phone 24/7, you can easily transfer funds online and the interest rates on savings and CDs are higher than brick and mortar banks.
If you're not ready to wholly commit to an online bank, set up a savings account, money market or CDs and keep your checking account. You can transfer money to and from all your accounts online.
Always make sure the bank you choose is FDIC insured.
Note: some credit cards allow you to get cash back when shopping. This might be an alternative to taking cash out of an ATM.
Two well known online banks are Discover Bank and HCBC (ING Direct who advertised a lot in the past is now owned by Capital One).
Go to www.Bankrate.com for information on banking institutions.