We always talk about using a local bank on this sub, but how can we evaluate if it's a bank worth working with?

I just moved to the Florida panhandle, and I have no network here. I am trying to find a local bank for an investment, following the advice on this sub. For example, this bank:

http://www.fnbt.com/

is local, has physical branches, but their website looks like it was put together 20 years ago. They are also not using website security (https), so I don't see going to them.

How do you suggest we determine if a bank is legitimate enough to work with?