Facility capture is a bit wacky. Unless I'm mad though, all there is to grasp:
Small bases have one to three control points. Stand near a control point long enough, and you capture it. Own the majority of control points, and you start to push up your score in the base. Once you've got the max score, you win the base. You can push the base more quickly if you have people standing beside each (now captured) control point, with the bonus per control point maxing out with some number of players.
Larger bases, such as biolabs, have sub-bases around the perimeter of their territory. If you capture a sub-base (same rules as any other base, though usually they have only one control point), you get a forward spawn point for you team to attack the main base.
Larger bases also often have generators. Shutting down a generator (hold E on it to plant a charge; shooting it doesn't do much or anything) will take down a forcefield, making it easier to enter the base. Some sub-bases -- e.g. biolabs -- have teleporters (icon is a little teleporter with up/down arrows) that beam you right into the main base, which of course makes tit much easier to take.
While you're taking a base, the enemies will generally still be able to spawn. The building they spawn in has team-pass forcefields that they can not only pass through but can shoot through as well. This is meant to discourage spawn camping, as well as give a team a chance to fight back if someone tries to snipe their base from under them. It's a bit annoying once it's become obvious that you've won though. Just don't stand in the line-of-fire, and if you're defending, give up and spawn somewhere else.
Bases capture more or less quickly depending on your influence on the base (how many hexes surrounding this base's territory that your team owns). I believe this is shown in the map.
They do a pretty poor job explaining any of this, so, eh. It could benefit from some better in-game explanation.
As far as figuring out how to get into the action: if you don't have anyone else to play with, just do the auto-squad-join, and press Insert to Instant Action to your squad leader. Whenever you need to respawn, the Green spawn point in the deploy screen is the one that's closest your squad leader. The game is definitely better if you have someone to play with, though. I probably wouldn't play it lone wolf.