For me, Shinjuku ftw, We stayed in Tokyu Stay Yotsuya for 14 nights. Larger than usual hotel rooms, has a kitchenette and washing mc which is a bonus! No room service and room cleaning is once a week. There is a station (Yotsuya) nearby, 5 minutes by foot. Four lines pass through this station so it is very convenient. It is away from the busy tourist areas but enough restaurants, snack and convenient stores. Shinjuku main station is 3 stops away, but be aware that it is a very huge multi-level station (reputed to be the largest in the world).  It is always busy, rush hour is a nightmare. We got to Hakone using Odakyu from Shinjuku station. We also did the so called Hakone Loop which covers Gora, Mt Fuji, Lake Ashi etc. Mt Fuji is really worth it and should be considered as part of your trip to Hakone. Make sure the weather is nice, i.e. not cloudly or worst rainy. Mt Fuji can be very shy,lol.

Shibuya, I find more suited for young crowd and a bit chaotic as it is close to Harajuku (another 'young' ppl place). Tokyo station is a bit out of the way but most Shinkansen leave from here!

To be fair, any of the areas you mention is really good and as long as you are near the metro, everything else is ok. Tokyo, I think has one of the best of not the best metro network. Take a little time to study the rail map and you will be ok. The three lines I use most of the time to travel around Tokyo are the Red, Marunochi Line, the Yello Ginza line and the very useful but busy Green Yamanote Loop. Good luck!

We booked the Hakone Enoshima 3 day pass and stayed a night in Gora Shinanoki Ichinoyu which has a private onsen. Well worth it and we were glad we stayed the night as on the first day we did not get to see Mt Fuji. On the second day the weather was gorgeous and we had the full view of magnificient Mt Fuji, really a sight to behold. The best view was on the Ropeway and  at the Owakudani stopover. We didn't do the 'pirate ship' the second time as its just a bit gimmicky unless you want to visit the towns by the lake. After the second round to see Mt Fuji we returned to Hakone enroute to Kamakura and Enoshima. Both nice places to visit. Then back to Shinjuku on a local express train. Sit on the left side of the train to get a last view of Mt Fuji as the train leaves Odawara (I think!) From Hakone, its just another hour or so to Kamakura then another short journey to Enoshima. The trains there are quite unique. Google to find out more 

Maybe karuizawa? I got no experience there but from google seems doable as day trip. If you willing to go a little further and stay overnight like in Nagano / Niigata / Yamagata areas then I got more personal experience recommendations.