Use potent materials -- you need 800 str for max speed, 800 mag for max stamina, and 5000 HP for max RP.
For Silver, infuse with level 1 Pulse Gladiator for second wind, level 1 Cactuar for sprinter, and level 24 Microchu for dark horse, marathoner and health nut (this one will stick your betting odds at 2.2 for virtually every race). Always set strategy as the fourth chocobo head (stay behind the pack), and use boost when Blue Streak activates.
For Gold, infuse a level 1 Metalligantuar instead of the Pulse Gladiator, then do the rest. Set strategy as the second chocobo head.
There's a few other useful abilities but you'll never need them.