I find that credit unions are usually best for basic checking. Their fees are much lower than commercials banks'. They belong to ATM networks with good availability, including internationally, and many will reimburse fees charged by the ATM networks. Years ago credit unions all had very narrow restrictions on who could join but today they're much more open.

The answer changes a bit if you have a significant financial portfolio. The big banks often give great features when you combine multiple account types with them-- checking, savings, brokerage, credit card, home mortgage, etc.