Hi again!
I'd be more than happy to give you some recommendation but please let me know your final selection of places to visit. This way I can give you the best recommendations!
Here are some to start with.
Given that you have 10 days. Start in Tokyo, two days. Start with the local high lights, Asakusa (for the temple), Ginza, Tokyo Tower, Odaiba (lots of big malls and entertainment for the kids), Harajuku and Shibuya for fasion, the the Shibuya crossing and the statue of Hachiko.
Then on to Kyoto, my recommended visits are the bamboo grove and monkey park at Arashiyama, the endless Torigates of Fushimi Inari and Golden Pavilion and Kyomizu-dera temple. Two or three days is what I would at least recommend for Kyoto.
Osaka, the most lively atmosphere is found near Dotonbori/Shinsai bashi. Here all kinds of small shops can be found selling food and local wares. It's a great place to try out Takoyaki. This is also close to the Namba area where brand stores and bigger department stores can be found, such as Bic Camera and the Apple store. Other great places to visit in Osaka are Osaka castle, Shinsekai and Universal studio. You could do 1-2 days for Osaka.
Then day trip to Kobe, visit to local China town the experience the atmosphere and try the food. Then to the harbour land for shopping and strolling around. You can have dinner trying Kobe Beef.
On to Hiroshima for a day. Visit the peace park and the island of Miyajima. Famous in Hiroshima is the local Okonomiyaki and Hiroshima Ramen style. Both are well worth trying.
Then back to Tokyo for 1 more day before departure home. You could also use this day for last minute shopping. Visit Shinjuku, as you'll find any kind of store there and you can combine it with a visit to the Shinjuku metropolitan building from where you can have an amazing skyline view of Tokyo for free.
Hope this helps,
Daniel