Red zones are off-limits to the public and heavily patrolled. Getting into an extended firefight with a squad of soldiers will just lead to disaster when an airship starts honking its fog horn and tanks roll in. So to get to objectives, you have a few options: drive a motorcycle as fast as you can, carefully sneak past each patrol, or just run.
I recommend the latter. The motorcycles are hard to control and likely to throw you off or get stuck in something. Carefully sneaking around is alright for a little while, but gets tedious when you realize how much of the red zones you have to traverse. Running is surprisingly effective. You’ll be spotted, probably multiple times, but if you just keep running you’ll probably get where you’re going mostly unharmed, with no one tailing you. To some degree, this is a flaw of Homefront, almost cheat-like and probably not the intended solution. But I’d rather be accomplishing missions than getting into long futile scraps on the way to each target.