If I buy a new domain name (an old one, already indexed in google) and redirect my current website to that one, will it get banned for getting a lot of links so fast?
No, because it's technically not getting all those links, but the redirection passes the PR to the new- bought (old) one. But that will only last if both domains focus on the same theme otherwise the PR will vanish after a while and you have to start (almost) from 0 again.