For a pleasant drive out of Tokyo, eschew the expressway for the more scenic route of the Koshu Kaido (Route 20).
Assuming you have the time to do the slower drive - the Chuo Expressway is generally much faster - Route 20 takes you through the countryside past Mount Takao. If you don't want to keep on driving you can stop at Mt. Takao and either hike up the mountain or ride the cable car up and then take a short walk along the concrete path. At the top you'll have a view of Mt. Fuji if the day is clear and you can have a light lunch of noodles or snacks. If you're hiking wear sturdy shoes as you'll have to climb dirt paths. There are three trails to choose from and each takes about 90 minutes. If you're taking the cable car you can even wear high heels as many of the young Japanese ladies seem to do (not advised, but you can if you want to). Parking is available for a fee, usually 500 yen, at the base of the cable car area and there are several restaurants and souvenir shops there too.
From Mt. Takao the highway takes you through a few twists and turns and after a few kilometers more you'll come to Lake Sagami. This lake was created in 1947 by damming the Sagami river. It was also the site of the canoeing events during the Tokyo summer Olympics in 1964. If you want to venture out onto the lake you can rent a rowboat. Fishing is also available with bass the most common fish.
If you're still in the driving mode, shortly after Lake Sagami make a left and head for the mountain village of Doshimura. This area of Yamanashi Prefecture is dotted with many "auto-camps" (camp grounds) where fishermen abound in the spring when the trout are hungry & the fishing is good. On our last drive on this route we stopped for lunch at a place with architecture that looked like a miniature English village set into the forest near Lake Yamanaka.
It's called Daieika - Big English House. Food wise, nothing spectacular, but "okay" omrice (omelet rice) swimming in a house made demi-glace sauce and a wafer thin, crispy crust pizza; enough to keep us going till evening.
Daieika is a busier place in the summer season and according to the owner's son, who was our host & chef for our stop there, his father bought an old building along the highway & started transforming it into this bit of England transplanted to Japan. There is an authentic enough looking pub area, but we settled for a walk-through and a return to the dining area where there was a friendly explanation about how the place came about. Seems his father is a builder, with a few hotels and other places in his experience book, and this place continues to be a bit of a work in progress. His brother is a wood craftsman & makes fish models, which are for sale in the little shop in the front along with some other items that may or may not be genuine antiques.
Leaving Daieika we headed on over to Lake Yamanaka, which is the biggest of the Fuji 5 lakes. It seems to be a Mecca for bird watchers with cameras at the ready. Of course, there are also the ubiquitous swan boats if you're looking for a self-powered cruise on the lake. We stopped to watch the bird watchers and enjoy a cup of coffee from the thermos while the water lapped the shore. Actually, pretty relaxing - a "zen" sort of feeling.