Classic mode consists of 6 maps, each with 10 challenges. For each map, there are:

3 high score challenges, where you must score above a specific amount of points total in one session (2 minutes)
1 high combo challenge, where you must score above a specific amount of points in a single combo
Collect SKATE, where you must collect the letters S-K-A-T-E that are scattered around the map during the course of a single session
Collect COMBO, where you must get the letters C-O-M-B-O in a single combo
4 other goals, which vary from map to map.
You do not have to get all 10 goals in a single session. Also, you do not have to collect either SKATE or COMBO in order, but for COMBO it is easiest to start at C and go in order.