Free: Contests & Raffles.
I'd go the route through the corner of Utah. The other way is going through Jackson and Grand Teton National Park, which is not a good way to go if you're pulling a trailer. It's very steep and mountainous. I'd only go that way for the scenery. The other route would be much better with your trailer.
From Western Washington I always go 90 all the way to 25 and then south to Casper. I'd be surprised if there is a faster route.
If a guy lives close to I-84, I sure wouldn't drive north to I-90. That can be a nightmare, just getting to I-90.
Really depends on your starting point, however it is hard to beat I90 to I25 from almost anywhere in the state. Montana is much more relaxed on speed enforcement than Oregon and Idaho IMHO, there's something to be said for setting the cruise control at 85 and crossing 555 miles of Montana in 7 hours with one stop for fuel and potty breaks. Have made that drive dozens of times. However, if you must start from I84, take I80 to Rawlins and 287 to Casper. Pass weather would be my determining factor in choice of routes.