For this achievement you need to win at least one race in all offline modes .
These modes are:
More details below but important to note despite winning all these game modes multiple times in career mode the achievement didn't unlock for me untill I got my first Gold time in Time Trial mode.
Anyway, if you are going for all the other achievements in this game you will win these naturally during the many cup completitions in career mode as they all appear multiple times.
If you just want to bang it out you can select each type individually in Offline>Custom Race.
Then select Papaya Grove has a short course called Avacado Roller that's less than a minute long. Stick to the red strips where you can for the most speed, green is average speed and blue slows you down so only cross it when you have to and quickly.
You select the Game Mode after you select your course, then set the race to Easy and the number of racers to 3.
If you have another controller you can join the race as a local second player and this will eliminate a random AI driver and it's associated lucky item wins designed to punish you at the last minute, basically leaving you to race one computer controlled driver instead of two. As a bonus you'll unlock: