Beginner and Rally are parts of Simulation Mode, so they don't have anything to do with it.
Go to Arcade Mode, then Single Race, then pick a difficulty. You see a bunch of tracks listed across the bottom:
Super Speedway
Mid-Field Raceway
Smokey Mountain
Swiss Alps
Trial Mountain
Mid-Field Raceway II
And in the corner you see Area A. That means that all 6 of the above tracks are in Area A. You need to race each one of them at least once and get 1st place, then you will open up Area B, which adds to the list of tracks:
Smokey Mountain II
Tokyo Rt. 246
Grand Valley Speedway
Laguna Seca Raceway
Rome Circuit
Tahiti Circuit
Once you win each of those races, you'll unlock 6 more tracks in area C, and so on, until you get all 34 tracks.
Go to the Clear Status section and it'll show you you all the areas and which races you've completed.
At first it's a little confusing when you can only see the first 6 tracks, because the "Area A" in the corner doesn't really make sense. I never even went to the Clear Status section for a long time because I thought "Clear Status" meant "Reset Arcade Mode".