Doug, this in the "Beyond Europe" section of the forum. : )
Martina, Gardiner entrance is the lowest portion of the park. It's bit more "scrub-brushy" compared to the forests, meadows and rolling plains you'll find deeper inside the park/caldera. It's not a bad place for a base for forays into the northern portions of the park, but compared to what you'll see inside the park, it's not really a place I'd spend too much time "exploring". West Yellowstone is touristy and not what I'd call the best national park experience. I get that it has lots of lodging, and that's why people stay there, but if you can find lodging anyplace else, you might want to consider skipping that area. But, be warned. The park is huge, speed limit is 45 mph, so you WILL be spending decent amounts of time in the car if you want to see the cool stuff.
Obviously, Mammoth Hot Springs is the closest major point of interest to Gardiner. Worth a few hours to explore. But while you're in the north portion of the park, you really need to spend time in the Lamar Valley. That's where you'll see lots of critters, especially if you drive through there early in the morning or later in the afternoon/evening. Bison, pronghorn, both kinds of bears, wolves, and more. It's the area they call "America's Serengeti". Also, check out Roosevelt Lodge. There are cabins there, albeit a bit rustic. But it positions you closer to more of the more interesting areas of the park than Gardiner. Also a nice place to get dinner, or just relax on the porch of the main lodge with a beverage watching the afternoon colors change.
It gets much less crowded as you go east from there toward Cooke City. Exit the park through the northeast entrance and you can drive up Beartooth pass (about 11,000 ft!), with great hiking options along the way, or head toward Cody via the Sunlight Basin, which is also a gorgeous drive. If you want to get to get back to the heart of the park, drive from Roosevelt over Dunraven Pass, stopping at the lookouts along the way for glimpses of wolves and bear. That takes you to the Canyon area, where you'll find more crowds, but also opportunities to hike and see the falls and canyon. Really spectacular. If you can get lodging here, you'll be in an even more central spot to see the rest of the park.
Following the Yellowstone River toward Fishing Bridge, you'll drive through the Hayden Valley, where you'll see lots of buffalo. The Yellowstone Hotel area is nice and central to everything if you're staying there, but can be a bit crowded and less "wild" than what you may be looking for in a National Park experience. Obviously, the geyser basins are a major draw. Probably less people and more interesting hikes in the Lower Basin. But the Upper Basin has the iconic Old Faithful and a few others that are much easier to access. Driving toward the south entrance, you'll see less iconic Yellowstone sites, but also encounter less people.
Some people, after a few days in Yellowstone, want some variety of scenery. If that's the case, you could exit the south entrance, and head toward the Tetons, which I've seen dozens of times and still never disappoints.
Finally, Bozeman is a really cool town, with great restaurants and a fun vibe. If you can, try to spend some time there (as opposed to just making it your departure point).