How do I wire a remote fan switch and still be able to use wall switches?
Get a different remote and receiver
If you want a remote and receiver that will play nicely with having light switches in the wall that work, you'll need to set what came with your fan aside and get something else instead.  My suggestion would be to get a Lutron Caseta fan controller and matching dimmer; they install instead of your existing wall switches and provide both wall and remote control.