For both versions - floor it! You need to be going as fast as possible at all times to stay away from the photographers chasing you.
For a regular Escort mission you'll need to make stops at XXX theatres, kill specific targets, cause traffic damage, powerslide, stop completely, perform near misses and rack up air time to please your passenger. Keep dodging into side streets where you can but don't leave the district in case you need to get back to a porn theatre.
For Tiger Escorts you need to keep your speed up at all times or kitty will lash out and cause you to swerve wildly across the road. The animal activists only make it madder but speeding up and getting away from them will make it's rage meter drop again. If you can, stick to the main roads and bridges and just be careful of any traffic accidents.