what would you recommend?

Depends how much time you have, and what you want to do. Most tourists try to visit Tokyo and Kyoto, perhaps with a few side trips. For a 10 day (9 night) trip, I'd suggest:

5 days based in Tokyo:

Day 1: Tour Shibuya-Shinjuku area. Start in Shibuya, see the Hachiko statue and walk up the shopping street. See Yoyogi Park and Meiji Shrine. End your day with some shopping and dinner in Shinjuku.

Day 2: Tour northeast Tokyo -- Asakusa temple, Ueno Park / zoo, river cruise on Sumida River

Day 3: Tour central Tokyo -- Imperial Palace, shopping in Ginza, museums

Days 4-5: Try some day trips. Suggestions are Hakone/Mt. Fuji; Nikko; or Kamakura/Yokohama/Minato Mirai.

4 days based in Kyoto:

Day 1: East Kyoto -- Ginkakuji, Philosopher's Path, Kiyomizu Temple

Day 2: West Kyoto -- Kinkakuji, Ryoanji, Arashiyama

Day 3: Central Kyoto -- Nijo Castle, Train Station (cool architecture), Pontocho walking street

Day 4: Day trip to Nara to see the Buddha and deer

Happy travels!

Go to the temples, they're beautiful. If it's possible, go to Kyoto, it's very traditional and extremely beautiful and interesting. Visit the temple Kiyomizu dera.

Mt Fuji!