Here are some things you want to look at for evaluating a bank or credit union for your regular spending accounts:
Convenience. Do they have a branch in a convenient location for you? Do they have no-fee ATMs near you?
Website. If you are like me, you will spend more time on the bank's website than you do inside a branch. Some bank's websites are great, some are terrible. Unfortunately, this is generally difficult to evaluate until you actually get an account. You want a website that is easy to use. It should allow you to easily move money between your accounts, get instant lists of transactions, show you your monthly statements, and have a billpay feature that works well. If you use budgeting software that interfaces online with your bank, you want to ensure that it works well with your bank.
Fee structure. Some banks will nickel-and-dime you to death. Watch out for minimum balance fees and ATM fees. Banks and credit unions usually have a fee schedule page on their website that lists every fee they charge, making it easy to compare different banks.
I would not be very concerned about interest rates for savings. Currently, all savings accounts have a universally terrible interest rate. Therefore, I wouldn't base my bank choice on the interest rate. Sure, one might offer double the interest rate of another, but double "next-to-nothing" is still "next-to-nothing." When you accumulate enough savings that you want to start maximizing your earnings, you can look for a better rate at another bank to move your savings to, and you can keep your checking account at the bank with the best convenience and fee structure.
In my limited experience, I have had better luck with credit unions than with banks when it comes to fees.