For Asia- Thailand, Malaysia, Cambodia, China. I have a group of friends who had a really horrible experience in Vietnam, so for that reason I am not recommending it.
As for traveling with tour groups, it's honestly not that hard to get around on your own. Plus you'll likely only see the really touristy places. Some of the more touristy places can be like Disney World in the summer- just packed to the gills with huge tour groups. If you really have no idea what to see in a city, there are lots of very short organized tours you can do. If you do a bigger organized tour, you won't get that much free time at all, most likely.