The first chance you will get to do 'vehicle' challenges will more than likely be once you purchase the horse from the toy store. When saddled up, the race start points will be lit up ready for you to go and start the race.
If you are struggling to get a gold medal on the horse riding races, then I personally found the achievement easier to get once you go into the Enchanted Glen.
Once you get on the dragon, coin collection races will start. The first of these races is the easiest and involves jumping around the map collecting coins. Collect all the coins in the time allowed to get gold.