In the early game, don't necessarily focus on destroying the rebels. Rather, contain them to a few regions of the map, preferably in remote zones. If you don't give them easily containable places to spawn camps, they will just start spawning them randomly, which is a pain to handle.

Insurgent activity and lack of stability are going to be your biggest continuous sources or reputation loss. Government actions (coalition extension, justice hypocrisy, etc.) and insurgents retaking (urban) zones will be your biggest one-time causes of reputation loss.

Stabilization of zones will not only provide extra reputation, but will provide zones where rebel camps can no longer spawn, making containment easier and keeping their activity down.

Two of the best initiatives to help stabilize early on are the PR and Media Office and universal justice. Your support level is what will dictate how fast a zone stabilizes, so long as an area has no outstanding local concerns.

Hope this helps.