Races are scattered through the city - they are marked on your map with a stopwatch-looking icon.
There are 15 races total, but for this achievement you only need to complete 5. HOWEVER, in order to unlock all the suits, you need to complete all 15 (the Hornet suit is unlocked after your 15th race is complete).
Races are broken up into two categories: Insanely easy, and moderately easy.
INSANELY EASY RACES: These are the races that have nothing but the green rings to zip through. You can actually web rush the ring's center (RB) so these are seriously child's play. If you want to be certain to get these done with no hassle, you can spam the Web Rush ability (Hold RB), but most of the time all you need to do is tap RB and you will have time to spare.
MODERATELY EASY RACES: These can be a bit trickier, but not if you utilize Web Rush (Hold RB). These are the races that have Red and Blue rings to fly through: Red rings do not allow you to zip through them (RB), but Blue rings do. You can make these much, much easier just spamming Web Rush (Hold RB). Spamming that ability will literally slow down the clock so you can aim through the Red rings accordingly. Then you can just zip to the Blue rings when available. These may take a run to just get the hang of it.
Questions, corrections and neg vote justifications below.