So after the stuff hits the fan, get on that turret and start mowing down anything not a human. The majority of this ride is you attacking enemies on the opposing rig, but at a few points some enemies will climb on your rig, but you can hit them from the turret or your friends slice them up for you; just keep an eye on the area under you. So you can basically just keep looking to the right and shooting in the general area of the other rig's deck. When you get near the bridge, shoot at the rig's driver and that's that.
This next bit I believe is mostly for show, but stay on the turret and shoot any Brumak in the head/gun area. Drones will be on the ground, more Brumaks will appear, and a few Corpsers toward the end that I don't think do anything to you. The last Brumak does require your attention, but you just aim the turret at his head and fire.