Before you do anything else, be sure to get the basic FCP X training from Ripple Training or macProVideo (who I produce for). I also have a course "Media Management In FCP X". Not vital, but could help in the long run when you have the time and money.
So, directly to your question, and a really good one at that. You get another Mac, first plug in that TimeMachine drive. Then boot it up for the first time. It'll ask you if you want to restore from a backup or TimeMachine drive. Say yes, point to the correct drive, done. Simple as that.
Everything will be where it should be. No worries.