All of the information in this thread so far is fantastic. I'll add my $0.02 since I've been here and done this.
Ours is a 6x14 tandem axle with a 7 foot ceiling height. I definitely recommend the tandem axle if you can swing it, for a variety of reasons, many of which have been mentioned above. If at all possible, get a 7 foot ceiling height. It's the most annoying thing ever to have to duck, or lower something.
Our trailer has "surge" brakes (self-contained hydraulic brake system) that works
okay. They take a second to kick in sometimes, and there's still plenty of load and strain put on the tow vehicle's own brakes. You definitely need some kind of trailer brakes. We went with the hydraulic surge brakes, which is kind of like a master cylinder activated off the trailer hitch, so that we could tow it with any vehicle, and didn't have to mess with any wiring within the tow vehicle.
I find the 6x14 to be the perfect size. We've moved a lot of things in which a 12' length just wouldn't have quite done it for us. The width is manageable. I would not be comfortable driving anything wider. I've been through some narrow lanes. Also, the 6x14 is really a breeze to drive with some practice. With the right tow vehicle, sometimes it's like it isn't even there. And, you can get it and your tow vehicle into two tandem parking spaces, without sticking out into an aisle. I've put about 14,000 miles on ours, including a trip from Southern California to Indianapolis and back (through the Eisenhower summit, fully loaded with 2 FRC teams worth of stuff).
You need a jack that doesn't suck. I'm currently using
this one. The wimpy ~3k lb top wind we had to begin with barely had enough travel, and looked like it was going to buckle over at any moment. You need to size the jack for the height of your trailer and vehicle.
We laminated FRP (yes, like in the 09 game) to plywood to install as interior paneling to our trailer, so we have more than just the thin sheet metal skin it came with, and that has worked well, and it looks a lot nicer than just bare plywood.
As stated above, do not underestimate the need for a stout (and reliable) tow rig. You do not want to tow at or near the capacity of your tow vehicle. Trust me on that. Also, Chris's comment about the combined gross vehicular weight rating is spot-on, and often neglected when it is actually very important. When looking at your CGVWR you need to look at all the people and stuff in the tow vehicle, in addition to whatever you're towing. And people are heavy!
We are towing with a 96 Suburban C2500 (3/4 ton) with the Vortec 454 7.4L engine and 4L80E trans. I picked it up for $2200 with 99k miles, in need of the front seats to be recovered, hood painted, and an AC system leak. What was really attractive was a had whole tow package. On top of the already heavier 3/4 ton leaf springs, it had load-leveling air bags, a hitch, and a trans cooler. We got it to specifically not put the wear on our own personal vehicles (Ranger and Ridgeline) which were undersized for the task. Suburban 1500s (1/2 ton) with the 350 and 4L60E can be had for even cheaper. We looked for an F250 or F350 or similar truck for over a year, and could not come up with anything in our budget, that was not all beat to hell or had over a quarter million miles. That's when we had the idea to start looking at SUVs.
If at all possible, you'll want something with a factory tow package that includes the proper frame-mounted hitch, and (very importantly) a transmission cooler.
Our 96 Suburban C2500 has served the purpose very well, but not without its faults. The Vortec 454 runs like a champ, smooth and pulls hard, but man is it a thirsty engine. We got 6.5 MPG coming back from Chezy Champs a couple weeks ago. Best we've ever done (no towing) is about 10 MPG. I averaged about 7-8 MPG out to Indianapolis and back, when our trailer was lighter, before the plywood went in. Also, the distributor gear that mates with the camshaft shredded all its teeth in the middle of a corn field in Iowa in July, 2013, and we managed to cook the trans on it coming back from Chezy Champs 2015, to the point where it needed a new trans. We also had to put a fuel pump in it on that same trip, at a cost of $1600. Apparently that's very common around the 100k mile mark on mid-late 90s suburbans. We recently did a new AC compressor and condenser in it too, and we did new shocks a while back. So, we've definitely put some dollars into it. But, it still comes out cheaper than a new Suburban, and it's just a great vehicle. It was quite the pinnacle of 1990s GM engineering.
Towing up a 6% grade, I still wish I had even more power sometimes. While I've never weighed it, I figure our cargo easily weighs over 3,000 pounds, and the trailer itself is probably up around 3,000 pound as well. Then, add 1200+ lbs of people inside the Suburban. With that load, I'm cruising about 50 MPH up a grade, even with the big-block tugging it, unless I want to drop it into 2nd, and be pushing about 3800k RPM the whole way up, which isn't great either.
Everyone knows the towing speed limit is 55MPH, but try going 55MPH up I-5, and you'll have miles of cars lined up behind you, and many angry folks passing you. On I-5 through the California central valley, 70 is more the norm towing, closer to 80 if not towing. You don't want to be stuck with a vehicle where you have to keep the tach pegged in a low gear to hold 65 when towing.
A diesel Ford Expedition would be the hot ticket, but I'd settle for a V10, or even an E-350. I wouldn't mind a newer 8.1L Suburban either. Plus, vehicles like Suburbans, Excursions, or 3/4 ton vans have the added utility of being able to carry 7+ passengers, which has been a real asset to our program.
I would never tow with a mini van. The suspension and especially transmission are just not up to it. You need something with a real frame, a real solid rear axle, a trans mounted long ways, not sideways, and some real displacement under the hood.
Remember, with the acquisition of a trailer, your program will grow, and you'll find more things (and weight) you'll want to put in it.