April could be dodgy or great. Do not know where you are located, will assume the TC area. You will not find much work except day labor so budget from there.
Head out 169 to Marshall, pick up I 80. First night out figure Chamberlin SD. State park Campground is pretty decent, KOA has hot water.
Day 2 should put you around Sheridan, WY. Badlands are in between. Now is the point of decision over the Bighorns through Medicine Wheel or up to Miles City and Custers Last Stand. Neither is a bad choice, let the weather decide for you.
From there it is open country and an adventure. The Worland Basin is great, I90 through Billings is also good. The passes through Yellowstone will probably be open so that is a thought. Especially the Madison River Country up through the Bitterroots.
Sunlight Basin and Dead Indian Trail are also an interesting way. Do note beyond Cook City the devastation of the Forest Fire are evident. You may want to head up over Beartooth Pass. If the snow has cleared hike up to the Rifle Pits Chief Joseph dug to cover his Winter withdrawal from General Crook.
Decision time Idaho by the Interstate or the 2 laners. Keep a weather eye open for late snowstorms, I have personally seen the passes closed in May because of a late snowfall.
In this area there are plenty of good Mom & Pop non chain restaurants. These are cheaper and better if you run across a 4B's or Black Angus the dinner specials are great and inexpensive.
Plan on State Park or National Park Campgrounds as a cheep place to stay. Seattle will be expensive and points going south. You are covering a lot of territory.
PS: I grew up in Park County WY on a working Ranch. There are seasonal jobs in the area however you are coming between seasons.
On the rest of your list I find standard and boring. Not take the two laners up from Albuquerque to Colorado Springs. Go through Santa Fe and Taos, From Colorado Springs take the interstates through Denver and across Nebraska. The original Cabelas is worth a stop.
There are campgrounds on your whole trip. Enjoy the adventure