Imho the lever that needs to be pulled first is internalizing the costs of parking space. Free parking in cities must go. A parking spot should cost about as much as a small apartment. That's because it consumes about as much space as a small apartment if you put a couple of stories of housing above it. Abundant parking aggravates sprawl and makes all forms of transportation more expensive.
Tokyo is already ahead in this. There are basically no surface parking spots available in the inner city.