If I get a new computer will I be able to play the Beta on it after I already played it on a previous computer?
[noob]
Do I have to buy another copy of the game?
If you look at Elite: Dangerous as past precedent you can install that on as many machines as you like but you need to log in to your Frontier Store account via the launcher in order to start the game and only one instance of the game can be running at any one time.
We'll have to wait and see on Planet Coaster but I'd imagine it'll be similar.