How do I get past Week 3 on Horsez?

What you do is go for a stroll and it will tell you to do a certain task and once you finish that your popularity will go up a bit

Chatroom:
Once you have gone into the Chatroom once and the Principal says you must spend more time on other subjects, you need to go for a Stroll and help other people find things they have lost, collect flowers for their girlfriends... Etc. Once you come back form the Stroll you will find that your Popularity has gone up a little. You may have to go for several Strolls before you fill up your Popularity.

Race with Antoine:
For the race with Antoine, there is no need to worry, I didn't beat him (and I don't know anyone who has) and I still completed the Adventure. Antoine gets expelled later on for stealing Clarisse's camera though he gets in a very annoyed with the Principal for expelling him as he says that he should also expel Ginger (your character) for taking part in the race (the Principal eventually finds out about it and tells her (and probably Antoine too) that if either of them breaks the Academy rules again, then they will be expelled and Antoine did, though Ginger didn't).

Hope this helps and I'm sorry if it is too long or not very clear (or both),

Kimberlie

Do stroll a lot and you will interact with others and raise your level

Just stroll around. If you do that your popularity will go up. Even if they don't give you a task to do, and you just get to ride freely, your popularity will go up.