For me, sidemissions have a somewhat insignificant impact on the citizenry's support. What won me my first town was assaulting a nearby airport and blowing up all their aircraft and armor there. Like has been said in other threads, power plants are key. They will win over every town they power soon enough. Holding nearby points of interests (like an outpost) for good and repelling counter-attacks has a significant and immediate impact on civilians' support. I don't believe stealing their cars lowers their support, but I could be wrong. If it does, it's pretty inconsequential.

Overall, keep in mind, Antistasi is the mod you have to stay in for the long haul to really accomplish anything.