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As far as routing goes, I'd suggest that you consider using some secondary roads since you're in no great hurry. Some of these are scenic in their own right, but also get you away from the main roads and Interstates which tend to get all the traffic in these sparsely populated areas. Specifically I'd recommend US-16 west out of the Rapid City area to just west of New Castle WY. Then switch over to WY-450 through Thunder Basin National Grassland. Jog right on WY-59 to WY-387 west to Midwest WY where you can continue on WY-387 or switch over to WY-259 (my preference) to connect with I-25 south.
At Casper the easiest route to follow would be to take Exit 288B onto Poplar Street (WY-220 west) and take that to US-287 south to Rawlins. Rawlins is a fair bit closer to the halfway point than Casper. Rawlins is also only about one-fourth the size of Casper, so you might find it a bit less hectic for an overnight stay. Day 2 would then just be I-80 to Salt Lake City.
As for places to stop along the route(s) suggested above would, on Day 1, include Bessemer Bend National Historic Site at Casper, as well as Independence Rock state Historic Site, Devils Gate and Martin's Cove (all on WY-220 near the junction with US-287). Then on Day 2 consider Flaming Gorge National Recreation Area, Fort Bridger State Historic Site, and possibly Devils Slide, a short detour west on I-84.
AZBuck