Which ever bank you decide on, watch out for mothly service fees. When we started looking most wanted to charge a monthly service fee on our checking account unless we kept a pretty high minimum monthly balance. We went with Wells Fargo because they wave the fee if we have an automatic monthly transer to an associated savings account and a minimum of 10 transactions per month on the debit card. Since our "business" is owning a few rental properties, it's easy to get the 10 transactions. About a year after opening the accounts they started sending us pre-approved offers for a business credit card with no annual fee and a little cash back on purchases. Will probably apply as soon as I get my FICO up a little.
Keeping your business and personal accounts separate is a great idea, but they don't need to be in different banks IMO.