Step one was to get the old chain replaced with the new one. The manual recommends that you break the chain, hook the new chain to the old, and pull it through the sprockets; I did that, and the hard part was keeping the new chain off the floor.
I had run the adjustment bolts most of the way in, and as it turned out I had run them in just far enough that I could connect the ends of the new chain with the provided link. This tells me that the old chain was seriously worn and stretched and in dire need of replacement. Well, if it was original equipment, it had 14k miles on it, and I was flirting with the final hash mark in the adjustment indicator.
Getting the new chain's tension dialed in was, of course, a bitch and a half. What I need is a handy tool with a magnetic mount and a couple of movable pointers on a millimeter scale, something I can adjust the angle of so that I can get it perpendicular to the chain.
Instead I had to hold a set of calipers. I checked the tension, set it, checked it again, made some minor adjustments, checked it again, tightened the jam nuts and the axle nut; then I checked the tension, loosened the axle nut, loosened the jam nuts,m reset the tension, checked it, tightened the jam nuts, checked the tension, tightened the axle nut....
After all that, I started the bike and ran through the gears with it on the center stand. Shutting off the engine with the rear wheel still spinning, I heard no unusual noises. A bit of a buzz from the chain going through the sprockets, which is entirely normal. Everything looked good, so I ran a bit of 10w-30 onto the chain and spun it by hand a few times, then put the chain guard back on and ran the thing up to about 40 MPH or so. The buzz was lessened, and everything was hunky-dory, so I called it a day.
...I called it a day, and didn't work on the Jeep, because right in the middle of the whole chain replacement task it decided to rain rather heavily, and if I don't have to lay in a wet driveway I'm not about to.
Just now, I realized that I forgot to put the cotter pin back in, so I'm going to go do that now.