Bully

The Go Kart Races are unlocked when the Carnival is unlocked in Chapter 2.

There are 5 which need to be completed, and you'll need to do these to unlock the Street Go Kart Races (required for another Trophy)

Just enter the Carnival and follow the Trophy symbol, it will cost you $1 to attempt each race.
When you win one race, you'll unlock the next one.
You win 2 carnival tickets for each race you win also, a small bonus.

If you are decent at racing games in general then this shouldn't be too hard because the go karts aren't that fast and they have decent grip.

The races get more twisty and turn heavy as they go along and adds more competitors (up to 6).
The best way to win is to get ahead as quickly as possible in the starting straight.
Avoid collisions with other go karts because this slows you down and you'll most likely end up getting sent head first into a wall.
The powerslide is your friend (X and Square together).

There's also one turn which you can cut across in various races (look out for the lowered rumble strip), it also makes it easier to do the next turn if you cut up this corner. It first appears in the 2nd race and your opponents will abuse this shortcut so make sure you do.

One thing which is most annoying about these races is looking at the map will not help you determine what kind of corner is coming up because it shows all the possible roads which are used throughout the 5 races.
So keep your eye on the road and not the map (except to gauge how close your opponents are behind you).