Wow...three days in Yellowstone, huh? Most definitely only enough time to barely scratch the surface, but a whole lot better than no time at all!
Since you've never been there, here's what I'd suggest for a three day itinerary (given your accommodations):
1. Day One-Old Faithful. Watch her blow two or three times-once right in front of the Inn with the masses of other people, another from up above on a short hike toward Solitary Geyser(where you may be alone), and then a third while you explore the area of the geyser basin that's accessible via boardwalks. If you start really early, this can be comfortably accomplished by lunchtime. After lunch, I'd really recommend getting away from OF Inn, and maybe doing the Lone Star Geyser trail. Check it out
here .
2. Day Two-This is a tough day because of the travel involved, so we'll have to use the travel to your advantage! You'll see a lot of the park on this day, but won't have time to get in-depth anywhere. On the way north out of OF, stop at Fountain Paint Pots trail, and do that (maybe 45 minutes total). If you're interested (and it's a highlight for my family), you can stop for a swim at the Firehole River swimming area on Firehole Drive, just before you get to Madison. Continue on to Mammoth, and allot at least three hours to tour the terraces. Again, if you're interested, there is a bathing/soaking area near the sign for the 45th parallel just north very near Mammoth. It's one of my favorite places to relax and hang out in the Park, but the last time we were there, the area was closed
. Continue over to Canyon before sunset for the scenery on the way.
3. Day Three-Again, up early-maybe even really really early to try and spot some wildlife in the Lamar Valley. It'll be very easy to spend the rest of the day on trails around the Canyon. If you're up for it, try Uncle Tom's to the bottom. Just remember that it takes twice as long to climb back out!! Make sure you hit all of the scenic viewpoints on both the north and south rims, especially (IMO) Artist Point (Hey, this IS the Dis, right? Gotta see the restaurant's namesake!!).
4. Day Four-Yes, there's a day FOUR. You have to get to the Tetons somehow, right???? And that's going to take you right through the Lake area. On the way, check out Sulphur Caldron, Mud Volcano, and one of my favorite areas, the LeHardy Rapids. Stop for breakfast at the Lake Hotel, then continue south. Consider another short stop at the West Thumb Geyser Basin for a short walk around the boardwalk. Watch for fish warming themselves over the submerged hot springs. Then continue south to the Tetons.....
Yes, you'll be exhausted. And yes, it's a lot to try and fit into a relatively short time. But it will give you an excellent overview of quite a few things, and gets you away from the crowds in a couple of places. Yellowstone is truly a classic National Park in every respect, and one thing holds true there that is true of virtually every park I've visited; All you have to do is get one mile from a parking lot, and you'll be virtually alone-no matter what time of year you visit. We've had a number of friends visit Yellowstone in recent years, and that's the most important bit of advice we've given them (and they've all confirmed was the most important thing!)-get away from the crowds and experience Yellowstone in solitude. It's an extraordinary place that you'll remember forever!!
Enjoy it no matter what you choose to do!!