When it comes time for a mortgage, shop, shop, shop. Terms vary greatly, & you want to deal with a mortgage company that will close in a reasonable time. Your credit score matters. Your bank doesn't.
For a checking account, look for zero fees, free bill pay, free checks, free mobile deposit, free ATM/debit card, online access, FDIC. The bank can be online or storefront. We have two checking accounts: Schwab, which does all of this & also handles our brokerage accounts, & Wells Fargo, which serves as a firewall account for occasional PayPal purchases, checks to individuals, & any company that doesn't accept a credit card payment. Zero fees can sometimes be accomplished by direct deposit of a minimum amount, or by also opening a savings account with an auto transfer every month. A storefront bank can be useful if you need services like a certified check or deposit lots of checks.