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Ah, in this case, here's what I've got for you.

Я буду играть первым. Играем по очереди. Одну гонку я, одну ты.

One thing though, it's hard to say why exactly, but по очереди после... This...

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...