Bicycles are different from cars, so cyclists need their own rules. This could be very simple — bicycles yield.
Cyclists would treat stop signs and stop lights like yield signs and yellow lights, slowing down to see what's coming and proceeding if safe. When two cyclists meet, the cyclist on the right would have the right of way.
Cyclists could also use sidewalks without disturbing pedestrians. Instead of passing a pedestrian, cyclists would dismount or move to the street, or even coast at a walking pace. If we want to be specific, we could forbid cycling within 30 or 50 feet of a pedestrian, but "no passing" would work as well. This would be safer for cyclists and better for traffic flow.
Let's adopt "bicycles yield" — why not?