Here are the basic things you want from your bank:
-- a local branch that's close to you and good customer service at the local branch;
-- free checking and saving account with online access : basically every bank offers that;
-- good car loan and mortgage rate. For this, you either go for credit union or national bank;
-- at least one of your bank should provide financial adversary service and wealth management;
-- if you travel a lot, you want to have world-wide ATM service with min fee.
That's pretty much I can say about the bank. If you find a local branch that you are comfortable with, you probably will stay with it for a long time.