Who throws first on each tee and in the fairway?

Under rule 802.02 (Order of Play):

• First tee: the order players are listed on the scorecard.
• Every later tee: lowest score on the previous hole throws first, and so on. Ties don't change the order.
• After everyone has teed off: the player farthest from the target ("away") throws next. If lies are the same distance, the lie established first goes first.
• If you're re-throwing from the same lie, you stay next.

To keep things moving, you can throw out of turn if the player who is next agrees or won't be affected. Otherwise throwing out of order is a courtesy violation (warning first, then 1 penalty throw for each later courtesy violation in the round). Don't walk past the away player while they're throwing.