You pretty much nailed most of your trip down with your comments. Since you are staying only 3 days in Tokyo, the rest of your time (4-7 days) should be in Kyoto. As much as it pains me to say so (being an Osakan), you would wasting your time going and staying in Osaka.

As for specific places to go, your question is too open-ended. I have no idea what your interests are, so I can't really confidently give you a list of places you ought to consider going to. Are you into art? Culture? History? Uniquely Japanese? How's your Japanese language skills? Or do you prefer the touristy trek? As Pickles noted, you haven't exactly asked any specific questions. I'm assuming a lot with the following.

I'm going to throw out a list of places that might appeal to all sorts of people. You will need to do your own homework to look up details and see if they're really places you want to go to. You didn't really expect us to do all the planning for you, did you?

In Tokyo, day one: Asakusa Kannon (Sensoji), Ueno Park and its assorted museums and zoo, Ame Yokocho, and choose one: Sumo museum; Jibocho, Imperial Palace East Gardens, Akibahara. Day two: river taxi from Asakusa to Hama Rikyu, Tsukiji, Edo-Tokyo Museum, Kabuki-za. Day three: Meiji Jinja, Harajuku, Roppongi Hills, Tokyo Metropolitan Government building, Ebisu Beer Museum (for a quick sit-down and good cheap beer at the end, not the displays!), window shopping at one of Shinjuku's department stores or a walk through Shinjuku Park.

In Kyoto, in no particular order: Kiyomizu-dera; Kinkakuji; Ginkakuji; Gion/Pontocho; Heian Jinja; depending on what special exhibits are going on during your visit, one of Kyoto's museums; Sanjusangendo; Fushimi Inari; one of Fushimi's sake breweries; Nijo Castle/Nijo Jinya; day trip to Kibune/Kurama; day trip to Himeji Castle and Kobe's Sannomiya (or if in season, a game at Koshien stadium in Nishinomiya); day trip to Nara; day trip to Arima hot springs and Osaka's Umeda; day trip to Mount Koya.

Again, with the places to stay, the question is too open-ended. What were you looking for? I got a ton of places I can recommend, but it really depends on how you want to be located (close to the sights, quiet neighborhood), budget ($50 tops? $100?), expectations (Japanese-style ryokan? Backpacker's hostel? Western-style bed?), among others. Just be aware that budget sleeps are quite rare unless you're willing to sacrifice something.

In Tokyo, you should look into the Taito ward hotels/hostels/ryokans. Two I can recommend are Ryokan Asakusa Shigetsu and Homeikan. I heard Sawanoya was good too, but I've never been there. Otherwise Sakura Hotel and K's Backpackers Hostel may be what you are looking for.

With Kyoto, I'm actually not too familiar with, since I never needed to find a bed there. I understand that there are not too many budget choices and they're all scattered around the city. This you will need to research on your own or hope another FTer can chime in.

Alternatives worth considering are Toyoko Inn chain or a homestay via JNTO. Toyoko Inn is like the Japanese version of Holiday Inn: won't win any style points, but the service is Japanese standard and you'll have a bed and hot shower for a decent price. A homestay's merits speak for themselves.

I'm sure other FTers can add some more to this and flesh it out. It behooves you to do some research on your own too, though.