I've seen several articles like this recent one at the Wall Street Journal, How to Pick a Bank. Most of them assume that you're shopping for one new bank. In my opinion, there's no reason to limit yourself to one bank. At minimum it makes sense to have two banks or credit unions: the first one with branches that are local and the second one that's an internet bank or credit union.
A local bank or credit union can be useful for making cash deposits and for services like safe deposit boxes. A branch that's conveniently located is one important consideration. However, for credit unions that are part of the shared branch network, that is less important. The shared branch network allows you to perform many banking transactions at a shared branch just as if it were your own credit union. However, shared branches have some limitations. For example, cash withdrawals may be limited to $500. So having your credit union conveniently located has some advantages.
The other important feature to look for at a local bank or credit union is a totally free checking account. You might first want to check if there are any local banks or credit unions offering a high-interest reward checking account. The vast majority of reward checking accounts are free with no monthly fees regardless of meeting the monthly requirements. So even if you don't care about interest rates, reward checking accounts should be considered.
Once you have a checking account at your local bank or credit union, it's easy to open an account at an internet bank. The internet bank will likely provide advantages over your local bank or credit union. One is higher interest rates for savings accounts and certificates of deposit. In addition to higher rates, they can give you more access to free ATMs. For example, at Ally Bank ATM fees charged by other banks are automatically refunded for any ATM worldwide. Other features that you will often find to be better at internet banks include online bill pay, bank-to-bank transfers and mobile banking. ING Direct and EverBank are two internet banks that offer mobile banking. EverBank also offers an online check deposit service in which you can scan your checks at home to make deposits without having to mail in the checks.
An internet bank doesn't necessarily have to be a bank. It could also be a credit union. One example is Alliant Credit Union. They have many branches in Illinois and in other states. However, Alliant has made it easy for people to join and open accounts online. As I described in my recent Alliant Credit Union review, they have many of the same features of the best internet banks. Also, anyone can join by an association membership.
If you were to limit yourself to only two banks, which would be your internet bank (or credit union)? Would the other one be a local institution? And what are the important services for your local institution?