Leaving Mammoth, I took a snow-coach to Old Faithful. Snow-coaches in Yellowstone come in two flavors. The classic variety are the purpose-built vehicles that featured in the old Bond movies, made by Bombadier as long ago as the 1950s, and still in service for Yellowstone tours. The more modern, but less romantic kind, are regular vans outfitted with tracks rather than tyres. They don't move very fast. From Mammoth, it takes 4 or 5 bumpy hours to make it to Old Faithful. Along the way, the driver generally makes a couple of scheduled stops at thermal areas or landmarks near the road, and also stops whenever interesting wildlife is sighted - eagles and bison on my trip. If you're familiar with the bus system in Denali National Park, in Alaska, the Yellowstone system in winter is very similar. Just as in Denali, it's the luck of the draw as to how knowledgeable or entertaining the driver is, but I was lucky twice so I surmise that you're unlikely to be disappointed. It's a memorable trip.
Biscuit Basin trail (5 miles out-and-back, flat). This is an easy trail which passes by a succession of geysers and other thermal features en route to Biscuit Basin. The only downside is that Biscuit Basin is on the main road, and frequented by parties of snowmobilers. But most of the time it's empty - quite a contrast to the summer crowds.
Black Sand Basin trail (4 miles, almost flat). This is another easy snow-shoe, which starts along the same route as the trail to Biscuit Basin. Once I left the main trail, I saw no-one else until I got to the geyser basin. The panorama above was taken at the Black Sand Basin, which on the day I visted was over-run with bison grazing near the warm river waters.
Lone Star Geyser trail (9 miles, 240 feet of elevation gain). A longer, but still fairly easy route up to a reliable geyser that's a bit off the beaten track (even in summer, never mind winter).
In addition to these easy trip - which were all I tried - the Park issues a winter trails guide that lists maybe half a dozen other moderate to more difficult trails. All told, there's certainly enough to keep one busy for several days in the vicinity of Old Faithful.
References
The National Park Service maintains a list of winter services in Yellowstone, along with a web page of businesses with permits to offer tours and other activities.