As someone who works in banking, I'd say it really depends on what you're looking for. Small and large are comparable when it comes to the benefits of a bank (i.e. free checking, assorted accounts, trust - not sure if you're looking for that, but hey, thought i'd throw it in). You may get a higher interest rate with a larger bank, but where they really differ is connivence and customer service. Smaller banks, in my opinion tend to have better service and are far more personable. For that I'd say go with a bank like Regions, Metabank I believe is quite popular, or any of the other small banks. The larger banks, you may run into some problems with service and if there is an issue with one of your accounts, it could end up being a huge headache. That being said, before I had my job, I used US Bank for 6 years without an issue. I've also heard a lot of good things about the service of Wells Fargo, but I haven't been a customer of them, so I can't really speak from experience.
Short version: I'd say pick a large bank, either US Bank or WF. To me, the benefits far outweigh the potential costs.