A few suggestions:
Overland safari from Maun to Kasane through Chobe National Park (Two to three nights) then from Kasane it is an easy 2 or so hour travel to Livingston.(via taxi (shared for private) or minibus)
Second, Many safari/tour operates offer transfers from Maun to Kasane and vise versa through Nata (Tarred road and easy travel the whole way). 65$USD more or less. In Kasane you can also get transfers to Livingstone (40usd) through lodges or tour operators. Thirdly, Bus from Maun to Nata at 0730 then bus from Nata to Kasane at 1130 (unfortunately you can not reserve seats and they don't always run on time.) We haven't been on a bus yet so I can speak of the chickenness of them, but in general, hey look quite nice and a huge step up from the rest of Africa.
We camped at Chobe Safari Lodge that had many tour offerings but also spoke to a guy at Dream Safari in Kasane. As far as operators in Maun, there are plenty.
Hope this helps!