Downshifting - you gotta know it
Avoiding hills is simply not an option.
Mountain driving is a lot like a roller coaster. The truck works really hard going up a long, long uphill, and then on the long downhill, gravity takes it and it goes like a rocket. Mountain driving is all about controlling this.
On mountain driving, especially with trucks, you will often find yourself needing brakes just to avoid exceeding the speed limit. This is a huge problem.
"Why?", you say. "I can push the brake all day." Maybe, but it won't work for long. Brakes are made for stopping at red lights, not descending long grades, and they will overheat, warp (which means a $400 bill on a car replacing them), fade (meaning they will lose braking power) and stop working, and you're a juggernaut. Happens all the time with trucks, and that's why they have runaway ramps.
If a cop catches you smoking a truck's brakes down a grade, he'll pull you over and find a way to stop you from driving further. For him, it's easier than spending all night at a fatal accident.
Enter the gears you never use on your shifter - P R N D 3 2 1, those. You have noticed when you lift off the gas on your car, the car "feels a little bit draggy", like it's actively being slowed down. If you select a lower gear (and you can select them while moving), it will become more draggy. Quite a lot more draggy the lower you go. That is for descending long hills.
The way I explain this is to downshift enough that you don't need to use your brakes at all. When you need to use a little bit of accelerator pedal to keep the vehicle at desired speed, you're in the zone. That keeps you safe.
Since you're trying to spin the engine to slow the car, this is a great time to blast the A/C on full, as this is "free" and won't overheat the engine.
Going uphill also has a trick
With modern automatic transmissions, it's easy - just put it in top gear and push the gas until you're at the top of the hill. (then downshift.)
However, there's a different gotcha -- this will make the engine much warmer than you're used to. Running this and air conditioning too can overheat your car/truck.
So you have to keep a close eye on your "Engine temp" gauge. First, you must know what "normal" looks like, so you must learn to watch it ordinarily. Then, on up-hill sections, keep an eye on it. If it climbs a little bit, no big deal, but if it climbs any more, turn off the A/C immediately -- most likely the overheat will cure itself.
In a pinch, it also helps to blast the heater and open windows (thanks Saaru). This heat comes from the engine and gives it a small bit of additional cooling. Don't get heatstroke!
Don't drive drowsy
Utah is the loudest about this, but other states agree: they really don't want you driving drowsy, and they hate cleaning up fatal accidents. If you are starting to nod off, pull over somewhere reasonable and safe, and take a catnap... often an hour will suffice, your body knows. Cops who see you asleep will be happy to see it.
Routes: the Union Pacific way
Generally any viable route is going to take you through Laramie, Wyoming. From there, does the vehicle handle wind well? If so, proceed west on I-80, which was built through a terrible wind area. Otherwise take US-30 west via Medicine Bow, paralleling the original Union Pacific Railroad and rejoining I-80 near Rawlins.
Wyoming is desolate, so be careful with fuel.
Continue west on I-80 via the Wasatch range, paralleling the Union Pacific the entire way. It's the best route, and it's flatter, which is why UP chose it.
Once in Utah, you'll hit a branch at Echo where I-84 turns northwest. Take it. It's your way anyway, and it's much flatter - still following Union Pacific. (Staying on I-80 will go an hour out of your way and put you on Parley's Summit, the worst grade on the entire 3000 mile length of I-80 - you certainly don't want to learn downshifting here!)
Congratulations, you have crossed the Rockies the easy way (the Union Pacific way).
From here, the only sensible road is I-84 west (actual northwest) - some time around Boise you should be planning your route into Oregon proper.