It sounds like you definitely want to go over your vehicle pretty thoroughly before you try towing anything. You really don't want to find out that something is faulty while you're towing, trust me when I say that is no fun and could result in injury or death. Don't be that guy.
You're on the right track, though. Just start running through the list. Check your gearing, as you planned.
Make sure the suspension is up to par. I'd skip the air/coil shocks, personally. They're a band-aid at very best, and a liability at worst should something fail while you're depending on them. Go with a good quality shock (Bilstein makes some great ones) and leave the load bearing to your springs, where it belongs. If your current springs are in good shape for normal cruising, I'd look at something like a load leveling system. Airbags assist is one decent option, as is something like a Hellwig spring assist system. I'm not sure what is available specifically for the wags, you'll have to look around a bit. If your springs needs a bit of assistance with no load, you could always drop in an add-a-leaf. The compromise there is a bit of unloaded ride quality. You're upping the static spring rate, so it's going to ride a little rougher without some weight on it. The load leveling systems are specifically designed to stay out of the mix *until* there is a load on the system, then they add spring rate to assist.
Brakes are one of, if not the, most important part of the equation here. There's a boatload of info on upgrading the brakes of these rigs floating around. If you want more assist, swap your vacuum booster out for a hydroboost (
http://www.ifsja.org/forums/vb/showthread.php?t=106056). For the front brakes, since you're planning to replace them anyway, go with the JB7 Chevy brakes (
http://www.ifsja.org/forums/vb/showthre ... =142737#15). There are probably other "big brake kits" available from big names like Baer, Brembo, etc. I did the basic JB7 upgrade on my J-10 and have been extremely happy there. For the rear, your only real upgrade option I'm aware of is a disk conversion. Some argue that it's not really much of an upgrade in terms of raw braking performance, but I absolutely abhor drum brakes. If yours are in good working order and adjusted correctly, though, they should serve you fine.
Go through everything in the steering, check it over thoroughly. Make sure all the tie rod ends are nice and tight and greased (if greasable). Make sure the balljoints are solid. Check the wheels bearings at all four corners. Check your spring and shackle bushings, track bar bushings, sway bar links, etc. Make sure everything is up to par.
Odds are the body mounts could stand replacing, they're 30 years old now. A lot of people have issues getting body mount bolts out, so start spraying them with a penetrating oil now. I'm a big fan of the 50/50 acetone/ATF mix, it's pretty amazing to see what that will do to a rusty bolt.
You can definitely add a brake controller yourself, if you want to put in a little effort. There's no magic to it, really. Whatever one you purchase should come with detailed instructions. I would recommend checking all your existing wiring as well. Do you have the relay box back by your hitch? You really don't want to start running trailer lighting off your primary taillight wiring.
A sway control hitch is a full system. It will use a standard 2" receiver hitch, but has additional pieces that attach on both sides (tow vehicle and trailer) to help keep the trailer behind you and not swinging side to side. They can get pricey, but SO worth it. It is not a pleasant experience to have a trailer doing a jig behind you and dragging your tow vehicle wherever it feels like across the highway. It is not something that has to come with the trailer, you can grab one and add it after the fact. A good tow place (like a U-Haul as Stuka mentioned) can help set you up with an appropriate hitch for your rig and trailer.
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edit: spelling
1983 J-10 - 4.6L(MPFI)/CS130D/Hydroboost/NV3550/D300/44/44/3.54/Disc-Disc/32s/42 gallon 'burb tank