If you are in an Agency, you can get your teammates to trade you cases that will move you up in the faction you are trying to get in good with. We do this when an agent wants to change factions.
Once you are +1 or better, doing little favors is the fastest way to get better.
I know you can be good with two factions, but I've never heard of anyone being in good with all three at one time. Jojo? How would that work, since you go down with one when you go up with another.