Foreword: the servers for this game in 2023 are HORRIBLE. It will likely take you many, many searches until you can connect to a match.
In depth guide for newcomers to the series. You cannot play Public Online Races until you unlock them through story progression.
To do this, start a new game and complete the first tutorial mission. After that you will go to sleep and start a new day, where you have a mandatory hacking side mission you need to complete.
It doesn't matter if you fail it or not, you have now met the qualifications for Public Online Races. You need a minimum of two accounts to start a match.
Press the Back Button to go into the Game menu and press the Right Bumper to go into Online Contracts.
From here, go down on the map until you see Public Online Races and select that. If you are double boxing this, you will need your dummy account to have the same amount of story progression as your main. Otherwise, your boosting partner will need the same amount of story progression.
Both accounts search and you'll get matched up in an online race. Win or lose it doesn't matter, but whoever needs the achievement MUST cross the finish line to get credit for a race. Do this nine more time and the Achievement will be yours!
If you happen to get stuck on an obstacle during the race, hold Y to respawn.