Ok first of all the multiplayer in this game is garbage and doesn't work properly. I've had multiple lobbies where everyone is ready with a check mark and the race wont start. Also if you have 6 people in the lobby and 1 person doesn't hit the ready button the race wont start which is stupid.
As for the best and fastest way to get this and all other multiplayer achievements is to get 1 friend and boost it in a lobby with just the two of you. Once you get the 25 races in each category you will be around 60 to 65% towards 200. I recommend just doing the shortest rally races after that so you can also work your way towards the rally fan achievement of 500.
The achievements for this game are not particularly hard, just long and time consuming.