Check out IACE travel. They do a great job and have a Chicago branch.
I completely agree with the prepaid sim card.
It does depend significantly on what kind of experience you want. If you want the young hip scene, stick to Tokyo or Osaka. If you want to experience some Japanese culture, Kyoto, Nara, and Hiroshima are great. 5-6 days would be about right if you were planning on going to Himeji, Nara, Kyoto, Osaka, Kurama and Hiroshima. However I am not sure if your ten days would be best spent consistently traveling and taking 1-2 hour trips. I think it would be pretty taxing getting around and you may not enjoy your Japanese experience as much. Also, these things would be much cheaper if you bought the railpass but not sure that's going to be possible for you given your timeframe. One way to get extensive amount of traveling in would be to use overnight busses, such as Willer express.