Kyoto
- Walk to the top of Fushimiinari.
- Kinkakuji (temple made of gold) is very touristy but it's worth seeing.
- Go to arashiyama. There's a bamboo forest, some nice temples and a monkey sanctuary. If you go there in early-mid afternoon you can feed them by hand, and the feeding area has an incredible view of the city.
- give yourself 3 - 4 days full days here. I wish I could have stayed longer.
Osaka:
- The castle is worth a visit.
- Definitely do a day trip to Nara.
- I didn't actually do much in Osaka apart from going out drinking...I hear the aquarium is good.
Went to Japan just a couple weeks ago and did all these aside from Nara. I second their recommendation.
If you're doing the Kyoto stuff (especially Fushimiinari and the monkey sanctuary) bring really comfortable walking/hiking shoes. Fushimiinari in particular is much larger than I expected. The monkey sanctuary is about a 15-20 minute walk uphill.
Pure Water temple (Kiyomizudera) is also a fantastic place to go in Kyoto. Just make sure that when you enter the temple proper (the 100yen special buddhist birth simulation thingy) you are not followed by loud high schoolers.
It's possible to do Golden Temple (Kinkakuji), Silver Temple (Ginkakuji), Pure Water Temple and Fushimiinari in one day, but I would recommend spacing out the big walks.
Osaka Castle is great, best of the castles I've been to aside from maybe Himeji (which I heard is currently under construction). Much more history than the other castles I saw.
Osaka Aquarium is also great, you can reach it via the subway + a short walk. They have a ton of really cool exhibits including a whale shark in one of the tanks and a manta ray / small shark petting zoo. Lots of cool jellyfish, otters, etc. as well as the standard fish. Tickets are about $20, which might seem expensive, but at that point a kilometer walk in a covered, air-conditioned area sounded great.
Osaka also has some good retro game shopping if you go to Den Den Town. The Super Potato there didn't have much (nicknamed it "subpar potato" later) but the shops across the street are good.
Other general advice:
Osaka can be done as a day trip (or multiple day trips) from Kyoto so you can use Kyoto as a base for both, it's about a 30-45 minute train ride at worst. Saves you booking more hotels and moving luggage around.
I've stayed at both K's House Kyoto and Tour Club Kyoto, K's House is cheaper and more hostel-y but they only have private rooms if you have a group if I recall correctly. Tour Club's a bit pricier (was $70/night for two) but a shorter walk from Kyoto Station.
If you go, search out public baths. They're a great way to relax after a long day of walking!