hello and welcome to the combine,
Teh first thing you could do is to leave the civic center. If you click on the position button in the members are you will then be able to move around inside the building and also to leave it and exit into the city.
But if you really want to get into the game quickly, meet some new friends and just get to know the combine I strongly recommend you to join a faction.
So think about what you want your character to be. Do he want to be a Imperial officer, a miner and so on. You probarbly already have a pretty good idea about that. Then you simply check out the faction pages for a suitable group and apply for membership. It can usually take up to 24 hours before being accepted into a group. During that time I think you should sign up on their site/forum. Check out the their site and the combine site as whole.
But it won't let me leave... there's only seven rooms in the place. One is called 'Starting Point', one is 'Hangar', and one is 'Control Center'. Whenever I try to move to a room that has a door leading into blackness, the arrow doesn't work.
You don’t use the arrows in that situation, in the travel menu should be a button saying Enter or Unboard or something like that, you press it and then choose to exit the civic center. This can only be done in one specific room but I can't remember the nameof the room, you'll have to waste a couple minutes looking around for it.