Once you have completed the main quest to meet the gang leaders you can then accept short missions from them.
These are like the repeatable faction ones from the main game and generally involve you killing a target, kidnapping someone or protecting gang members.
You may get offered jobs which conflict with your morals, such as abduct someone from Diamond City. If you do not want a task/don't like it just turn it down and talk to the leader again and they will offer another, randomly generated task. You can find out what's involved by asking what they want and who the target is.
As these take place in the Commonwealth and not Nuka-World it is best to accept one job from each leader simultaneously to cut down on load times. This way it's only 4 trips.
To keep it simple it's best to do this as soon as the game has you meeting all of the leaders.
From cnirnava:
"It should be noted that there is a confirmed glitch which will cause these quests not to trigger / respawn in the current version of the game (v 1.7.12.0.1).
These quests should be available immediately after turning in the one prior.
The glitch (which I can confirm for myself) has to do with certain companions being with you. I had Gage with me. I was able to get the first quest with each gang, but upon turning it in, none of the gang leaders would offer me another one. I proceeded through the storyline checking back after every few sub-quests to see if there was another trigger. Nothing.
If you are effected by this glitch all you need to do is dismiss your companion, fast travel to one of the gang headquarters, and speak with them again. They should offer up a quest then and repeat once you turn it in.
I'm just putting this as a comment instead of a solution because I am sure they will release a patch soon to fix this."