Nearly any credit union offers bill pay for free. BECU (if your in the Seattle area) has a nice mobile app and website that are at feature parity, with a great Zelle integration. Otherwise, First Tech FCU is the largest US Credit Union, and tends to be a fairly decent credit union.
First Tech is fine, they have branches all up and down the west coast, and offer free access to the Coop ATM Network (every 7-Eleven ATM, any other Credit Union ATM), ATM fee reimbursement, Online Bill Pay, a working Zelle integration (which doesn't support recurring payments yet unlike BECU :c) and other fringe features.
They're working on a new mobile app, but they really need to add Zelle support on their website IMO. Bill Pay and transaction history are available via the website.