In the beginning of the game, after you roll to see who goes first, are the battlefields revealed and then the winning player picks who goes first? Or what is the order of beginning a game?

According to the official rules, battlefields are revealed before determining turn order. The winning player of the "roll to go first" (or any other fair random method) then chooses whether they wish to play first or last.

Based on the Core Rules and official FAQs, here is the official Start of Game sequence:

1. Legend selection
2. Champion selection
3. Battlefield selection and placement: Battlefields are chosen and placed face-up in the Battlefield Zone simultaneously. They are public information once placed.
4. Determine turn order: Players use a fair random method (such as a dice roll) to designate the first player.
5. Choose play order: The designated player chooses whether they wish to go first or last.
6. Draw cards (each player draws 4).
7. Mulligans (starting with the player going first).

Important Clarification: As noted in the RiftJudge FAQs, it is a common error to hide battlefields until after determining turn order, but this is incorrect. Because battlefields are public information as soon as they are placed, they must be revealed before anyone makes the decision to play first or last. If you are playing a Best of 3 match, this procedure is repeated for every game, with the loser of the previous game being the one who makes the choice of play order (after battlefields are already revealed).