Mostly, I am a member of a ton of credit unions, many of which do not have branches in my area. They all have some sort of brick and mortar presence, but it may be 3K miles from me. I have one local (may add more since my super-convenient local just moved) for the odd cash deposit and such.
Credit unions are great. They tend to have no fees unless you really screw the pooch like overdraft a check or something, and even then it's usually lower than a big bank. They are all member-owned non-profits, you are the shareholder. Most of them have some sort of "field of membership" limitations on eligibility, but there are often easy ways around it.
For starters, I would recommend Alliant for its great savings rates and fast ACH. The savings rates are not too noticeable presently of course, but comparatively competitive, and in times of higher rates they have historically been extremely competitive. Also, Pentagon for its great mortgage products and rewards credit cards to rival any bank. Both have fields of membership that can be entered with a small, one-time charitable donation. ($10-15 or so.)