Definitely go to Kyoto if you're into history and culture - and go to Osaka because it's next door. The Shinkansen from Tokyo to Osaka takes about 3 hours - by local train from there to Kyoto is around 40 minutes.
When I went I stayed in a hotel in Osaka because it's cheaper and then on to Kyoto for the day. Check out kinkakuji (the golden temple) and kyomizudera (a temple sitting on the hillside on huge wooden stilts).
For onsen I'd recommend Hakone - about 1 hour away from Tokyo. You can also try some eggs boiled in volcanic water here - you need to take the cable car.
As to Beppu and Sapporo - I haven't been .... yet. Hopefully someone else can chime in
