How about some advice from a Wyoming native (who's stuck in Georgia)?
Can't help with "what spares" as I'm also of the preventative maintenance guy and haven't taken spares with me, ever. In the 3 times I've been stranded, it's always been electrical (loose ground to coils, dead regulator/rectifier on a motorcycle and an unknown/unfound fault when the dew point and temp cause every surface to collect moisture). Either your 4runner is ready or it isn't, only you know for sure.
But in Wyoming, be prepared for roads that can range from packed snow (no clue where "center" line is, you follow the side markers) to glare ice to bone dry. Chains are fine, though some of the newer options (snow socks) might be easier to use.. regardless, put those on Now, before you need to, just so you know how they fit when it's light, and you don't need them. If your tires aren't in the first 1/4 of their life, you're going to have Extra fun on any slick roads... 4wd doesn't help you turn or stop, just gets you going well enough to get you into trouble if you need to turn or stop. Best advice I can offer to a GA citizen... slow the heck down and only do 1 (one) input at a time. If you need to slow down for a corner, get the speed down before you even think about turning. Similarly, you want to speed up after that corner, finish the turn before touching the throttle (this is where 4wd can get you into trouble). If your tires are in the Last 1/4 of life, you are going to be miserable anytime the road conditions go bad. Seems like most folks here in GA start thinking about new tires once they have the cords showing across the full width...
Take a snatch-strap and know your anchor points. You might need them for someone else to help you, or to help someone else.
A good winter survival kit, enough for 3~4 days. I've seen a few storms over the years that just blow in, obliterate all vision and close the road down for a day or Maybe two. Might be 3 or 4 days till a plow comes by for the first time depending on where you are and how bad the storm is.
Mind your fuel, it's easy to find places where you can go ~120 miles without fuel stops. Further than that if you explore and get off the main highways/make the wrong turns at the wrong places. And don't expect to have cell service (Verizon has the best coverage, but AT&T has coverage where VZW doesn't in some cases).
No idea where you're headed, but I'll gladly put in a good word for Lander, WY. It's probably the best place I've lived, and my wife was military so I've lived a lot of places (plus, I was a nomad for a decade before meeting her). It's a Government town (local, state and fed) so a good many white collar jobs. Riverton (sister town, 24 miles away) is the blue-collar town with majority being in the oil industry, but there is some ag/farming/ranching in the area as well. Personally, I'm not a fan of the south half of the state, too flat, dry, windy.
When I head that direction, first day gets us to Kansas City and 2nd day into WY. Either the southern route into Cheyenne or the northern route into Buffalo/Cody. These are 'long' driving days though, hope you're young (or can trade off with your GF to catch naps).
Good luck, hope you enjoy yourself. If you do find yourself in Lander and feel like pizza, hit up Tonys on main. Lander or Riverton and feel like the best sub/soup you'll likely find, head for The Breadboard. You'll be blown away by the steak/primr rib at Svilliars in Hudson (between the 2 above towns), just ignore the 60y/o decor.
