Hi,
Tokyo has something for everyone; there is no one-size-fits-all itinerary. There are a few popular favorites I can suggest - the first is visiting Harajuku on a Sunday to see the youth with the bizarre fashions - something that happens every weekend for decades.
https://www.youtube.com/watch?v=3tY7jPFhTBM
You can see it together with Meiji Shrine and Yoyogi Park, 2 other nice places bundled together. Another place that every last tourist goes to see is the Sensoji temple in Asakusa.
https://www.youtube.com/watch?v=SwjnQDjwaPQ
Akihabara is the electronics and manga/anime shopping area - huge clusters if stores selling every electronic thing you can think of. In Shinjuku there are other stores (Bic Camera, Sakuraya, Yodobashi) with competitive prices however, but just seeing Akihabara is quite a sight.
But for a list of many places to consider, try:
The Sky Tree (if the weather is clear)
Sensoji Temple in Asakusa
Harajuku/Meiji Shrine on a Sunday to see the bizarre fashions.
Odaiba at night for a real trip.
See Love Hotel Hill in Dogenzaka at night for something uniquely Japanese.
Tsukiji Fish Market at the crack of dawn.
Akihabara for the electronics.
Shinjuku and Shibuya at night for the endless lights and neon.
Shinjuku Gyoen Park
Hamarikyu Gardens
Kabukicho for the seedy and sleazy side of Tokyo
Kappabashi for the mindblowing realistic plastic restaurant displays for sale
Rikugien Garden
Ginza for the overpriced shopping
Kiyosumi Garden
Yasukuni for a real experience of Nihonjinron
Tokyo Metropolitan Gov't Building in Shinjuku (better view from the top than the Tokyo Tower, and
all free!). Otherwise try Roppongi Hills.
Roppongi for the ex-pat environment
I suggest you browse these websites for more info and narrow down what you want:
http://www.gotokyo.org/en/index.html
http://www.jnto.go.jp/eng/location/regional/tokyo/index.html
http://www.ar.jal.com/world/en/guidetojapan/city_information/tyo/see/akasaka/
http://www.japan-guide.com/e/e2164.html
You are putting in a lot of time in Tokyo - to get around cheaply, look into a 3 day subway pass:
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03
I'd also suggest you get out of the city for a couple days for some good day trips - popular places are NIkko, Kamakura, and Hakone. There are day passes you can use if your JR Pass is expired:
http://www.japan-guide.com/e/e2357.html
Since you are already seeing Tokyo, you can largely skip Osaka during the day. But at night the city comes alive and has some great places to see, such as Dotonbori, plus the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are stunning:
https://www.youtube.com/watch?v=COJ_MeXbeow
https://www.youtube.com/watch?v=xb1wuyAGgyQ
https://www.youtube.com/watch?v=-AyUZmGbW8g
I noticed that you did omit Nara from your trip going south - it really is a must see place and worth a day. Himeji Castle is also worth going to see. For Hiroshima, be sure you also go see Miyajima, and if you still have time, Iwakuni is a lovely castle town with its iconic bridge:
https://www.youtube.com/watch?v=0COOZXi66mM
Lastly, you will be in Japan during the height of cherry blossoms - expect huge crowds at many places. But be sure to go see some of them because they are really special to look at for just a short time during the year. Some popular viewing places are:
Tokyo
http://www.japan-guide.com/e/e3050.html
Kyoto
http://www.japan-guide.com/e/e3951.html
Nara
http://www.japan-guide.com/e/e4117.html
Hiroshima
http://www.japan-guide.com/e/e3407.html