Above: Undine Falls is east of Mammoth Hot Springs in the northwest section of Yellowstone. Tower Falls, located further east, is close enough for a "same day" visit.
The Mammoth Area: Is near the parks northwest corner. It has lodging and a nice visitor center. Nearby is a store selling groceries and souvenirs. To the left of the store is a somewhat formal restaurant. On the opposite side of the building is a "fast food like" restaurant. A bit "pricey" (restaurants in national parks always are), I nonetheless enjoy eating on the fast food side and get myself "well stuffed" eating both a cheeseburger and veggie burger . In the early hours of the day, a breakfast combination platter is available on a small plate. If you are heading out on a long afternoon hike, or if you have a big appetite -- I suggest skipping the breakfast and waiting for lunch to be served. The breakfast combination platter won't hold you for very long. For me , a can of bean soup cooked on my propane stove is actually a better choice if not returning for several hours.
At Mammoth , large wildlife sometimes congregate in large open areas. The "Upper Terraces Scenic Drive" , one of the most spectacular areas of the park , features many hot springs and is only a mile or two down the road in the Mammoth area.
I am not impressed with the campground at Mammoth as it does not look natural. Because of this, I have never camped there. I like just about all other campgrounds in the park better than Mammoth. Most of the parks campgrounds are run by the same company or by one other company. They are all pretty nice.
An alternative campground and a Trail: To the south of Mammoth is a campground called Indian Springs. It is a quiet spot and is very relaxing. The Bunson Peak Trail, also south of the Mammoth area, rises about a thousand feet in elevation.It really is a great hike and isn't all that hard. The view at the top is absolutely magnificent !
The West Entrance: -- Is known as West Yellowstone. The town of West Yellowstone is just outside the park. It has a number of restaurants and motels. I suggest calling, or getting on the internet before arriving. This is one of the busiest National Parks in the country - if not the world , so availability is extremely "iffy at best" and improbable without an advance reservation. There is no visitor center adjacent to the West Yellowstone entrance once you have driven into the park. There are campgrounds (Madison, then Norris - a favorite of mine) between West Yellowstone and Grand Canyon Village to the east. After Madison, you can either continue east (to Grand Canyon Village) or turn south towards Old Faithful. Both areas have hotel rooms , nice visitor centers , restaurants, a "General Store" and Canyon Village, a nice campground. I like the campground at Grand Canyon Village. Grand Canyon Village even has a laundry mat. Mammoth, Grand Canyon Village, and the Old Faithful areas all have showers available that campground users (for a small fee) can use even though they are not renting a room. Much of what is mentioned above applies to other park entrance areas and locations but I am less familiar with them.
WILDLIFE: - The road between West Yellowstone and the Norris camp has a lot of large wildlife - mainly elk and buffalo. You will actually see many of these animals walking along the shoulder of the road. While they are fun to watch--do not get out of your vehicle or stretch your arms or hands out. They may well "charge toward you" . Buffalo seem to me to be as dangerous, if not more more dangerous than elk. If you stand too close they give you a look that seems to say "stay out of my way" and when too close - they seem to walk toward you as you are trying to back away. This happened to me one time -- even though I was at least "40 yards" away. Close does not have to mean very close . KEEP A SAFE DISTANCE FROM ALL LARGE WILDLIFE".
Near Old Faithful is a trail to "Lone Star Geyser". There are two different ways to hike to "Lone Star Geyser". The short route was , to me, rather dull and at times , I felt Like I was hiking on a flat mountainless trail in Florida. The longer route is much more scenic with mountains clearly visible . I highly recommend taking the extra time on this route as it is a lot of fun. Check with a park ranger at one of the visitor centers to be sure you take the slightly longer route. The Lone Star Geyser erupts every few hours . I was fortunate enough to see it erupt after waiting about 45 or 50 minutes after my arrival. Old Faithful: The Old Faithful geyser erupts many times a day. It is an easy walk from the hotels , gift shop, and restaurants.