Split/Second

Simply follow an opponent through a shortcut they have opened. You can increase the chances of getting this achievement by playing levels that feature more than one shortcut. Below is a list of all shortcuts. The number in brackets indicates how many shortcuts are available after route changers.

Downtown Central: 0 (0)
Airport Terminal: 0 (1)
Construction Site: 0 (0)
Ferry Wharf: 1 (0)
Port Bridge: 1 (2)
Airplane Graveyard: 2 (2)
Dry Docks: 0 (1)
Expressway: 0 (0)
Power Plant: 1 (1)
Storm Drain: 1 (1)
Canyon: 1 (2)

At first Airplane Graveyard might seem like the obvious choice, but Airplane Graveyard has only two laps whereas Port Bridge and Canyon have three. This means that while you have four potential places to get the achievement in Airplane Graveyard, you have five possible places to get it in Port Bridge and Canyon.

Cheapskate

To unlock the achievement you have to successfully use a shortcut that a rival has triggered. Similar to the Better Luck Next Time achievement but actually making it through. Seems to work better if you're ahead of the opponent that opened the shortcut.