For this I was able to do all 5 races listed in chapter 1. You will need 7500 splinters total.
I did the dunkle race to buy the first bird which costs 3000 splinters, may not be necessary to switch birds but this worked for me. You could just repeat the seachantress side quest or do the side quests at cartohom nearby which gets you to explore the map to get the splinters and you can get a lot by doing these whilst making progress towards other exploration achievements at the same time.
I then got an imperial trade permit from the imperial embassy (near redmouth forge) for 500 splinters.
I then flew to Imperial Harbour and bought 2 benthic blue mutagens for 2000 splinters each - this put me up to speed 4. These are permanent mutagens until you switch birds. This is fast enough to beat the races once you get used to the bird's movements.
The location of the races are (in order of difficulty from least to most in my opinion):
Dunkle
Sark's Hollow
Mawbridge
Port remit (this has the best bird in the game)
Imperial Peak
You can find the races by looking for the yellow fire icons on the map near these locations, which look a bit like exclamation marks where the top part is a flame.
The races start where you can see a flame, then you have to follow the flame under or over the gates to get to the end. The races will tell you how long you have taken for each gate but will only tell you if you made the target time or didn't at the end.
It doesn't feel like the races are meant to be part of the game design at first but after a few tries you should get fairly used to it. Remember to dive to build boost and boost straight at every opportunity but don't try to boost and steer or you'll barrel roll. Steering takes priority.
Good luck.