One basic advice I can give is: use your feet! :-)
The central (old) part of Cottbus is not very large, so you don't have to use public transportation, but if you really want to, you can use a bus or tram.
I enjoy it to just walk around in some small parks, along the Spree or through the centre. Although there might not be any outstanding highlights, you can see many nice buildings, churches and nice, green places to have a coffee, beer or ice-cream.
Special tip: On sundays, there is free entry to a small exhibition for paintings and photos at the northern end of the "Sprem" (the pedestrian precinct in the centre).