I'd try to go Spring as much as possible. Winter basically just ended AFAIK (it was still winter season in the Biwako Valley ski resort) and it's already getting warm in Osaka. + Late spring means cherry blossoms which are purrty.
Some sights (mostly tourist-y) you should try to hit:
Kinkaku-Ji - Purrty, north Kyoto.
Kiyomizu Dera - Purrty, main hall under renovation until 2016, east Kyoto. The road towards the temple and along the Gion District are nice to walk along, with a bunch of small shops, restaurants, etc. in old-timey cobblestone style. Very touristy, still fun though. Mochi here is pretty good. Also there are nice (if expensive) places to buy legitimate tea sets and good quality chopsticks.
Sanjusangen Do - Lots of cool statues of buddhist gods.
Kyoto-gosho - Could get some good scenic spots in the spring time.
All the above are typically covered in day and half-day tours. They're reachable using the Kyoto metro too, but Kinkaku-ji's a bit farther than most other spots, especially if you book a place near Kyoto Station.
My suggestion for tours: Sunrise Tours was reasonable (check Viator), they meet at the New Miyako Hotel close to the southern part of Kyoto Station.
If you do take a tour, I'd suggest going back to Gion and Kiyomizu (even if you don't go up), cause there's a lot to see in those walkways, including more residential zones which don't really cater to tourists. Also if you manage to hook up, there's a love hotel district close to what I believe to be Nishiraiin temple.
Other than that, there's Kyoto Tower just outside the station (north of Kyoto station), the manga museum which I've never been to, and what looks to be a highly recommended ramen shop west of Kyoto station. Also the malls in and near Kyoto station are nice for light shopping and souvenir buying.
Never tried it but you can try visiting Kyoto Animation's studio as well.
Nara, obviously. There are day trips to Nara, but you can get it done yourself just by taking the Kintetsu-Nara line. The park is an obvious place to go. Don't forget to bow to the deer, and try not to get raped
(no seriously, they bit me in the behind. :c)
Lotsa pretty pictures in Nara, but that's about it IIRC. Shopping near the station as usual.
Yamazaki. Not many people go here I reckon. Known pretty much solely for the Yamazaki Distillery. Being Scottish and a drinker, it'd be a disservice for you not to go. Tours are free and you get 2 glasses of Whisky Highballs (and an extra two neat if you prefer), but in English they're only audio guide tours. You follow a tour guide who speaks only Japanese, so if your moon's good enough, I'd suggest taking that tour. You have to reserve in advance by phone only, but it takes about 15 minutes to get their by rail from Kyoto (or 25 from Osaka) and it's a beautiful little town. Pics to follow if I don't get lazy.
Shiga Prefecture and Lake Biwa. Honestly can't tell you much about this, but we went north to Biwako Valley ski resort and it was rather fun if you're into that. If you go spring time it should be open for some good views and stuff. If you have the time, it's worth exploring at the very least.
This place is like the new Hong Kong for us Filipinos. Cheap ass shopping in the form of Rinku Premium Outlets, but that's not what you came for.
Osaka Castle's okay. Nice view of the city, some Tokugawa vs Hideyoshi history. TBH it's rather missable.
Universal Studio Osaka has some Japan-only 4D rides if you're interested. Evangelion, Shingeki no Kyojin, MonHunt, Biohazard/Resident Evil.
The real meat of Osaka though (IMO) is the food. Head down to Dotonbori for a lot of shopping and some good takoyaki (the shop's name is Kukuru, there's constantly a line of both tourist and local). It's nice to walk around the area and just take in the sights. A lot of ****ing people though. Aside from that, for Okonomiyaki, I recommend the wrong side of the tracks near Nishinakajimaminamigata Station. Lots of watering holes there that serve the stuff. Have it with mochi. Good ****.
For more anime-y and weeby ****, hit up Yodobashi Camera near Umeda station. First floor is phone and PC stuff. 2nd floor for videogames and camera stuff (the PSVita has a hell of a lot of Visual Novels). 3rd floor for figurines. There's a Jump Shop on the 5th floor if you like their stuff.
If you're interested, I can do something about Tokyo as well.