The first one is spot on.
The second is a mess, sorry. По очереди is what you need there. The thing is I kind of need to know what kind of game you have in mind there to give you a correct translation. Like in chess, when a player makes his move it's his ход or if you need a verb, then use ходить. If it's dice, then бросать (кости) and the list goes on... It's usually ходить though
E.g: В этой игре ходят по очереди. В этой игре (do whatever) по очереди.