Aki,
I'm not much into Onsens, so I don't personally know of any good ones close to the track, but I will consult my better half and get back to you.
I have to admit that I have never driven on the track myself
Mainly because I don't have a lot of confidence in my skills.
I've always wanted to try a more "open" track like Fuji or Tsukuba first, build some experience and then try the more technical Sports Land Yamanashi (SLY) circuit. Having an experienced driver there to give advice and pointers is a great idea. The shortness of the circuit means you'll be able to complete a lot of laps. Top speed is irrelevant, but ability to brake, corner and accelerate quickly before braking sharply for the next corner is.
For gas, there is a very small local gas station on the way to the circuit from the interchange. Aside from that if you turn left at the toll booths and head down the hill into Niraski there are about 3 gas station including a Jomo and Eneos. The drive from the circuit to Niraski town (food, supplies and gas) can be done in 10 minutes.
A word of caution: There is one Koban in the first village you pass though and occassionaly, more often than not on Sat/Sun, there will be a cop car patrolling. I guess the locals don't appreciate the noise of tuned cars/bikes rocketing through there village every weekend.
The mountain roads around the circuit are a lot like the circuit itself. They're not wide with long sweeping curves like those you find at the Hakone Skyline or turnpike, but narrower with tight hairpin bends. If you head past the track the road is pretty much empty. There isn't much around there apart from a golf course and a couple of small villages. Eventually you can drive all the way to Shosenkyo, a popular tourist destination (a big waterfall, some superb scenery, shops selling trinkets, small restaurants and a thousand tourists piled into tour buses)