Just pay attention to your driving when on/near the rez. Unemployment there is over 70%, and there is a high rate of alcoholism and drug use. The rez is dry, so they go to either Riverton or Lander to pick up booze. I think Riverton is a bigger draw to the Natives than Lander simply because Riverton has the only Walmart in the County. With 70+% not working there are many DUI arrest throughout the day.
Everyone has an opinion on bear defense (especially on the internet, with most arguing not even living in bear country), so figure out what works for you. I carry a Ruger 45 Colt revolver with 325 grain WFN bullets in a crossdraw holster. I'm comfortable with it. Griz's have been killed with 10mm, 41 mag, 44 mag, 45 Colt, 357 mag as far as handguns are considered, and there's plenty of occasions where bear spray has been effective too. The only thing I'll say is if you carry a big bore handgun, make sure your practiced with it. Best way to avoid bear encounters is to not walk around with your head up your butt, pay attention to the sign around you, and make a little noise when the terrain gets tight, and don't forget to look over your shoulder occasionally.
Moose...lol! Some of the funniest stories from the mountains are moose stories. 9.9/10 times it'll be a cow with a calf getting pissy and running someone up, or around, a tree. I'm sure it's happened, but I've not heard of anyone seriously injured or killed by one. There are wayyyy less moose around (at least up in Park County) since the wolves were introduced. I don't know about the populations in Fremont County.
When you get here folks will give you both good advice and bullshit advice about recreating in the mountains. You'll sort it out.