Should the trap be before or after vent?

Depends on your perspective what is before or after. If you're following the direction of the water flow, the trap needs to be before the vent. A water filled trap blocks bad smells/dangerous gasses from entering your home. Putting the vent closest to the sink would bypasses that block. It would also likely cause gurgling because of siphon action trying to pull the water out of your trap (the vent breaks the siphon because its easier to pull air down the vent than to remove the water from the trap)

Right = Sink - trap - vent - main drain stack