You change the weather (and resolve any other events) first, and then move. See the turn sequence on page 6 of the rules.
Nothing on page 6 leads to that conclusion. On the contrary, the individual turn order on page 6 says:
3. Play individual game turns in the active section. Each individual turn consists of:
- Pit-stop (optional)
- Draw 1-2 cards
- Play 1-2 cards (optional)
- Pay resources, perform checks and resolve events (the latter only in the advanced game)
On page 7 it states that any action listed on a card takes place when the card is played, but movement also occurs when the card is played. The rules further state that if two cards are played that both have checks, the checks can be performed in any order.