All they get from their parents is color (and not 100%, just high chance) and chance at one of the abilities (Gallop/Canter being the usual racing abilities) starting out learned. Stats always start at lowest and have to be worked up. I've got a black male and red female that are both good racers (one with Gallop/Treasure Hunter, other with Gallop/Canter), but you wouldn't get a blue with that. What do you mean by racing though? If it's just chocobuck farming you mean, then that's pretty easy to get with any good rider (maxed STR or close to it, and decent/good END).
Racing in the Chocobo Circuit, though, is a whole other issue. Even if you know exactly what you're doing raising one for that, it takes a ton of gil and a lot of time put in every single day that it's in the stables, since you have to buy prolly over a hundred stat upgrades with chocobucks (which are 80 chocobucks a piece if your chocobo isn't retired yet, and 100 if it is retired). It's definitely not something I'd recommend unless you really know what you're getting into and have the time/money to put into it.
For the record, my red chocobo is my racer for Chocobo Circuit. She's got maxed STR, END is barely at the 2nd highest level, DIS Average, and REC A bit deficient (damn close to Average). She's maxed out on her stats (in that if I were to try and raise any one stat, it would lower another stat). I do races with her every RL day, for the most part (obviously I miss some days when I either can't get on, or just don't feel like logging on). It's a decent source of income, but raising her was like leveling a craft to 100, and definitely not a process I'd ever wanna repeat.