Do you have Bank of America near you?
If so,
I would do BofA Core Checking (Which is what I personally use), combined with BankAmericard Travel Rewards (no annual fees & instead of most cards where you only get 1 pt/dollar, you get 1.5 pts/dollar).

If not,

I would do checking at Alliant Credit Union www.alliantcreditunion.org (It's an online credit union that pays .65% interest on your checking balance) & I would do Amex Platinum, with a backup Visa to use wherever Amex isn't accepted (Standard BankAmericard is a good one, or Alliant's Platinum Card).

Whatever you do, do not go with Chase. Visit www.chase-sucks.com for more information on why you shouldn't bank with Chase.

Hope I helped!