You'll love it here! Wyoming is any outdoor enthusiast's paradise. I live in Casper, which is windier than the Bighorn Basin, but folks aren't kidding about how strong it gets, especially in winter. Make sure you have a good 4-season tarp with doors that you can pitch very tightly to block the wind. A good windblock will go a long way in staying warm. If you don't have much mountain experience, I'd recommend finding someone who does and going with them a few times before going alone. And certainly go hiking in the cold a few times before trying to camp in it.
With regard to the emergency GPS unit mentioned earlier, I definitely recommend one. I prefer the "personal locator beacon" type rather than the SPOT type. PLB's cost more but there's no annual cost (subscription fee) and they use a more reliable signal method. You can't really do the "I'm ok" texts, but if you need it in an emergency you're more likely to be found. They're pricey for sure, but here in WY you can go for miles and miles and not see anybody or any sign of civilization, so I think they're worth it.
Casper is a big city (for Wyoming), so I'm not sure how smaller towns handle it, but the roads here are plowed or at least packed pretty early after a storm. I wouldn't worry too much about driving around town, although my 4x4 has come in handy a couple times this winter. Winter tires are helpful.
Oh, and you have to be very serious about bear bagging in the mountains west of the basin. Carry bear spray too. Those bears are BIG.
Sorry, I kinda went off topic a bit. But I hope it's helpful.
Good luck on the move! Wyoming is a friendly place.
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