Over 10 days, I'd be thinking about only 2 places. It's not the Amazing Race (unless you really want that). Here are three trips I would consider.

*South Africa. Cape Town, drive up the Garden Route if you like driving tours, and then a few days in Kruger national park.

*Seoul and Beijing. This might be a little more sight-seeing than you want. I think the best months for this are April and May. July and August are very humid and I would not suggest the summer.

*Istanbul (or Rome) and Kigali, Rwanda. You can get to Kigali from Istanbul once a week on Turkish, or most days on Ethiopian from Rome through Addis Ababa. Besides the 1994 thing, there are gorilla treks. Kigali is not 10 days by itself, though.