Check out this link for ways how to repair your standing.
Second, I wouldn’t sweat running storyline missions the derived standing hits are small. The big thing is to avoid missions that ask you to kill ships of opposing factions. Of course, this missions also drop tags. And since a lot of players avoid them, these tags tend to be worth more. Thus, some guys run those missions because they’re more profitable, and living with the standings hit.
What’s the best route to take? Meh. I say there’s better money to be made elsewhere that doesn’t come with borking your faction standings.