By "exposing a set", I assume you mean the move when you take a just-discarded tile to complete a set you have in hand. When that happens:
- You declare the match ("pong" if you're completing a 3-of-a-kind, "chow" if you're completing a run).
- You reveal the matching tiles from your hand.
- You take the tile the other player discarded and add it to your set.
- You discard a tile, and play continues to your right (so if the discard wasn't from the player to your left, it changes the play order).
You don't draw a new tile, because the tile you took counts as your draw - you should always have 13 tiles in hand, or 14 when it's your turn, not counting any bonus tiles. You can only call chow from the player to your left, i.e. the one you'd be playing after anyway, but you can call pong from any other player's discard.
If you're completing a 4-of-a-kind ("kong"), then it is slightly different. You reveal your 3 matching tiles, take the discarded tile, then you draw a tile from the back end of the wall and discard a tile. The fourth tile in a kong essentially becomes a bonus tile that doesn't count towards your 13 tiles in hand, so you have to replace it so that you can still have a complete hand.