I haven't done much with companions so I might try a couple, like Boone and Veronica and stuff. I know that to start their quests you have to gain so many "points" first. Do they have to actually be standing there or simply be an active companion, or does it depend on the trigger?

http://fallout.wikia.com/wiki/Fallout:_New_Vegas_companions

they have to be standing there for all compaion points

As far as I'm aware they actually have to be there as an active companion. Having said that some are a lot easier to trigger than others, and it also depends on the 'trigger' as you thought. One, you may just have to visit x, y & z places. Another you may have to do a, b & c in x, y & z places. Then there are some for whom you automatically get their 'quests' when you team up with them.

Some companion quests are triggered by the same side quests. the Stands the Grass, Whitewash and Repconn facilities/quests are key to at least 3 potential companions, and there are others too. In some cases, you only get one trigger per quest/location, some you get multiple triggers so you can go back with a different one.

So, as you are picking your mix, keep this in mind.

Now, the thing to remember is, you don't need to do the companion quests to have a companion. The companion quests just add depth and back story to the game but are not necessary to using a companion. I never did any of the companion quests, other than the ones to get them, in the first two playthroughs. This time through I am trying to do all the companion quests this play through and it is a bit tougher to pick and choose which companion to have along so I get all the needed points.

I have a follow-up question. If you do something with a companion that gets points with them and then release them, will you still have the points if you go back later and "rehire" them? I picked up Boone a bit too early and want to do Veronica's first since most of his are a little later in the game.