Learning to fix flats is a good idea; you might carry an electric compressor (and a way to hook it up) or bicycle pump, a spare tube (front size, can be stuffed into the rear in an emergency), and a patch kit (should you puncture the tube with the tire spoons you'll carry); a little vial of dish soap helps lubricate the tire bead on re-installation.
Here's a useful video on motorcycle tire-changing:
Here's a $ 10 (Target) compressor:
PAY ATTENTION to how the tire hub components go together, which spacers, speedometer mechanism, etc., goes where; a print of the hub fiches might be a good idea to carry on-board.
A Clymer service manual might be handy; look it over and scan-and-print a page or two to carry (e.g., electrical diagram; also available on the 'Net).
Carry spare fuses; learn where they are so you can change 'em (substituing a 15-amp fuse for the 10-amp OEM headlight fuse might be a good idea; little thing tends to blow when HIGH/LOW beams inadvertently engage simultaneously).
For trail survival, I'd recommend handguards (to protect clutch and brake levers; spare levers on board might be a good idea, also), nerf bars (to protect radiator, tank, and plastics), and a proper bash plate (i.e., aluminum, replacing the stock plastic skid plate).
There's no limit to the mods you might perform, and the accessories you might add; I've tried to distill things to essentials.
Start out with the bike in good shape, preventive maintenance (e.g., oil-and-filter changes, air cleaner cleaned and lubricated, etc.) performed, and you'll be fine, armed with the preparations mentioned.
(Some will insist upon upgrading your doohickey (if it hasn't been done); from Bayesian probability, on a bike as old as yours, I'd bet you survive without this modification, although it's sound maintenance/insurance.)
Regardless, get out and ride that KLR; your insurance (perhaps through your AMA membership) will give you a tow if you need it!