Favorite part of traveling with the team? I don't honestly know--the one year I was a student on 1293, we traveled separately. That general arrangement has followed me through three seasons as a college student (and a team change, to boot). The perils of living a half-mile from your one regional...
That said, I have racked up a few miles going to other competitions--I'm aiming to break the 20,000-mile career mark this off-season. I've gone to events by plane (Cal Games), train (Mission Mayhem, 1902's Vex event), and automobile (everything else), and I have to say that I much prefer not doing the drive myself. (That said, the budget generally doesn't allow for this...but the travel is a side benefit, more often than not.)
My favorite part of traveling is meeting all the folks that I wouldn't have met otherwise. Sure, the biggest names in the FIRST world come within four hours of me once a year at the Championship--but catching them at a regional or off-season event can be a far more rewarding experience. Whether it's talking with folks at 4:00 in the morning, rebooting an off-season event in twelve hours, getting creeped out standing next to 254's tetra arm*, or just taking humor in a
slightly screwy field barrier, there are many things that just can't quite be replicated anywhere but in their natural habitats.
*On the way from SFO to San Jose with Cory, we stopped at 254's facility to load some things into his SUV. Their 2005 arm (with or without a base, I can't remember) was fully upright, a good ten or twelve feet up, and didn't appear to be held in that state through any sort of locking device. Cory did tell me it was safe, but I didn't feel the need to press my luck nearly three thousand miles from home.