Let me be the first of many to recommend Schwab banking.
Their ATM card refunds all fees worldwide.
Their customer service has always been top-notch for us.
Online billpay, transfers, etc. have always worked for us (pushing and pulling money is free).
We have not used any wire transfers with Schwab, but I feel confident in recommending them for your online banking needs.
also, last year they had several bonuses available--Amex cards and DD for checking account. We pocketed $600 cash ;-)