The hardest part about visiting Yellowstone is finding a place to stay.
Are you talking about this summer? If so, you may or may not be able to get a cabin inside the national park. If you can, you'll find that the accomodations are decent, though very basic; but because you're paying for location, location, location, it'll be expensive. I want to say that we paid about $160/night for a room in the Canyon area, and it was not as nice as a typical Holiday Inn. It was clean and decent, but it was small and not particularly inviting. We did get a bear-shaped soap in the bathroom, which my youngest daughter refused to use; in fact, she still has that little bear "sleeping" on a special soap dish in her bathroom. (Worth $160/night? No. It's a soap.) It redeeming feature was its location: Right in the middle of Yellowstone. Everything was still a drive, but it wasn't a bad drive.
No matter which portion of the park you might stay in (or visit), it'll have a cafeteria and a lunch counter (inside the gift shop). The prices are a bit high, but -- again -- you're paying for location. Our food was good. You do not want to keep food in your own car; bears can get in.
Your other good choice is to stay in the town of West Yellowstone, which is literally adjacent to the western gate. Here you'll find hotels for about the same price, but they'll be more on the lines of Holiday Inns (and will include air conditioning, TV, pools, and laundry facilities). In the town you'll like the Grizzly and Wolf Center, the Imax theater, a small rodeo, and the little shops. There's a great pancake place -- something Bear. The problem with staying in West Yellowstone is the traffic. In summer, you will literally wait 45 minutes -an hour to get in the gate of the park. Seriously, you will not believe the traffic until you've seen it.
Which is better? I really don't think Yellowstone has a really good lodging choice -- and I'm not overly fussy. I appreciate basic accomodations; I just dislike paying a premium price for them. You won't be in a dirty place or an unsafe place, but you won't find a value hotel like you could elsewhere.
Other ways to save money:
Do not enter the park without a full tank of gas. They sell gas inside the park, but you'll pay a premium price. If the gas station's open.
Be a defensive driver. Traffic is very heavy inside the park, and people are crazy. Everyone's driving along and SUDDENLY everyone STOPS. Look around -- there's sure to be a bear cub or an eagle nest nearby.
The $25 entry fee covers everyone in your car for one week. One payment gets you into both Yellowstone and neighboring Grand Tetons.
Pack for both cold and warm weather. In July we had jeans and hiking boots, but only short-sleeved tee-shirts. In a cold snap I was forced to buy $50 sweatshirts for everyone. They were excellent quality, but still . . .
Bring all the OTC meds, bandaids, etc. that you're likely to need. These things are very expensive in the park stores. Don't forget bug spray; the flying no-see-ems in Grand Tetons just about did us in.
Take advantage of the free ranger-led programs. We've always enjoyed them. In both Yellowstone and Grand Tetons you have loads of choices: Some hikes, some dealing with geology, others with animal life or Indian history. You can view the choices online.
Inside the Grand Tetons, definitely do the Hidden Falls hike. It's at least a half-day thing. You have to take the ferry across the lake, which costs a few dollars. Pack lunch and water -- no facilities are available in the Hidden Falls area. The hike is moderately challenging, but soooo beautiful. At the top you'll be rewarded with an unparalleled view of the parks. You'll want to stay a while, even if it's just to lie down on the cold, cold granite after the strenuous hike. My favorite pictures of our entire trip were taken at the top of this mountain.
If you go to Cody, Wyoming, which is on the eastern side of Yellowstone, definitely visit the rodeo. We had a blast! I can't remember what the tickets cost, but it was well worth it. I've heard some people say it's a po-dunk little rodeo, but we have nothing here at home to which we can compare it, so we thought it was great. Go early so you can get a good seat! In Cody, eat at the Irma, but don't bother with the gun show (overly crowded, and you can't hear). Another vote for the Buffalo Bill museum -- it's actually a complex of five museums (five?).
Yellowstone is so diverse. It's hard to grasp 'til you've seen it. Within that relatively small space, you can see canyons, pine forests, geysers, waterfalls, and lakes. Just hiking and looking is an adventure. Do hit as many of the areas and museums as you can; they are so different.
Do not miss Uncle Tom's Trail.
Things to consider, though they aren't budget:
We loved the Yellowstone horseback-ride-cookout. It's an hour horseback ride out to a place in the middle of nowhere (you can also ride the wagon, if you want), and they serve a great steak meal. Then you ride back an hour. Really a fun evening.
We stayed in a very nice cabin inside the Grand Tetons (Signal Mountain Lodge, to be specific). 2 bedrooms/1 bath, it was spacious and comfortable, and in such a gorgeous location. It was something like $250/night, but I felt it was well worth the cost -- I can't say that about the cabins inside Yellowstone. Also, we 'specially liked the food at the adjacent Trapper Grill.
As a general rule of thumb, everything in the Grand Tetons costs more than everything in Yellowstone.
If you keep going farther south into the town of Jackson Hole, everything costs more still.
We did a float trip inside the Grand Tetons. Our guide was great. I read that you should choose the last trip of the evening. I don't know how it compared to the other timeslots, but we saw so much wildlife.
This may be a bit far afield for your trip, but if you're going to the Mt. Rushmore area, do not miss Custer State Park. Wow. It genuinely rivals the national parks. We did the buffalo safari, which was well worth the high price tag. Our guide was an older man who really knew his stuff. He took us into parts of the park not accessible to the public, and we learned so much about buffalo. He also instilled in us a healthy respect for those dangerous creatures! We didn't stay inside the park here, but the cabins looked very nice.
All around this area, you'll see buffalo roaming free. Sometimes they're literally in the parking lot. Be careful! Every year someone is killed by these massive wild creatures, and usually it's a person who was trying to get a picture. Keep your distance. Park far away and walk.