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I recommend that you choose an online bank that best serves your needs and lifestyle. Mine, for example, offers unlimited ATM reimbursement, but others will cap the reimbursement amount or restrict you to a (typically large) number of "free" ATMs. Those banks, however, may pay a higher level of interest than my bank. NerdWallet did an excellent job summarizing the best online checking accounts of 2016.
2. Consider a community or association bank or a credit union. There's really only one reason I can imagine "needing" a physical bank, and that's to deposit cash. (For our family, even that is a rarity; if we get a wad of cash, we'll just use it for expenses anticipated in our budget, such as groceries.)
Beyond daily banking, however, it can still be good to have a relationship with a local bank or credit union, because they also tend to offer higher rates on deposit products and lower rates on loans. You may also want use them for a financial-planning tool that I value very highly for unexpected opportunities or emergencies: an unused home-equity line of credit (preferably with a rate no higher than Prime plus one, no origination fees, no annual fees and no prepayment penalties).