I've done Wyoming/Colorado to Pennsylvania four times, Colorado to Edmonton, AB once, and several other trips > 1,500 miles. I've never had any mechanical problems on any of the trips (knock on wood) nor tire blowouts.
I'm usually one of the slowest vehicles on the road - I just pull over when safe if I have cars following on the two-lane roads.
I look over both tow vehicle and trailer at every gas stop. I always check the hitch after being parked somewhere, in case some joker pulled a pin on me.
All of this is pretty minor - the most important thing: remember to have fun. That means being flexible with schedules, not trying to do too much road in a day, and if you are tired, finding a place to stop to sleep.
I also try to make the first day a short day where I'll "camp" somewhere near a Wal-Mart so I can pick up what I inevitably forgot.