The most well known hot spring town near Tokyo is Hakone, only a couple hours south by train. It also has good access to clear views of Mt Fuji.
From Hakone Town, you can take a small train that heads up the mountain. It's an interesting train in that it reverses back and forth as it winds up.
The onsen hot springs and ryokans are spread out at different stations on the mountain. There aren't that many places to eat or buy necessities here, the majority are down in the town or at Gora station at the top end.
Also at Gora station is where you can catch a cable car higher up. At the end, you'll transfer onto the Hakone Ropeway that takes you from there. The ropeway brings you to Owakudani, an area with sulfur continuously rising out of the mountain. This area is famous for it's black eggs which are boiled in the naturally bubbling water. Definitely try one, it's good. I also tried wasabi ice cream from a shop there which had a tiny kick to it.
Getting back on the ropeway, you can head down to the Lake Ashi before getting on a boat that brings you to the Motohakone area. On the last leg of the ropeway and Motohakone, you can get a clear view of Mt Fuji.
This area also has Hakone Sekisho Checkpoint, a historic entrance point to old Edo. After taking a walk around the area, you can catch a street bus that will take you back to the onsen area.
Around Hakone town, there's a bunch of shops with traditional Japanese foods and snacks. One even had an outdoor fire with free pieces of fish where you could grill yourself to try. Tofu is what Hakone is known for however, and we went to one of the more well known ones:
Yuba is the top layers of tofu as it's being made. This place services yuba-don's, which is cooked yuba over rice. They have lunch sets which are basic but tasty, and if like tofu, this place is worth trying. Just a heads up, because it's well known, this place can get quite crowded so go early if you can (It opens at 11AM).