I recommend basing in Tokyo. Osaka is a fairly depressing city for the most part, save the Namba area.
If you can arrange it, try to fly to Tokyo, and back to Taiwan from Osaka.
After a few days in Tokyo, make your way to Osaka via Hakone (I second Rascal - this is a gorgeous area. Also, there is a guest house you can stay at at very affordable rates. I believe they’re called the Hakone Fuji Guest House or something similar. They have free hot springs. Oh, but I think the area will be COLD in winter). Don’t miss the Open Air Museum!
…and after Hakone you will definitely want to visit Kyoto and then Nara (doable as a day trip from Kyoto).
If you get the chance, spend a day in Kobe. A real gem of a place but not so hot on the tourist track. (kind of like Slovenia in Europe!) Kobe can be visited as a day trip from Osaka.
Maybe spend a day in Osaka (nice castle) and fly out.
By the way, Tokyo is a bit like Paris. It’s a place to wander around and soak in the atmosphere of the different districts. Shibuya and Shinjuku and fun (though be prepared for chaotic subway stations!). I also like Shimokitazawa. Great cozy area to wander the streets and check out the cool little shops. Plus of course there are gorgeous parks, including one in Ueno.
There are also several nice day trips from Tokyo, such as Yokohama.
Anyway, that’s what I’d do with my first 10 days in Japan (if I had them over again, knowing what I know now. )