Probably the best way to get a racing choco with traits (digging choco with traits is easy) is to first train up its physical stats to first class (walk as chick, carry packages as adolescent and adult), then switch to dig for treasure (which will quickly raise discernment and moderately lower endurance) until discernment is average.
Spam stories until it learns Gallop/Canter, then adjust care plan to fine-tune your end result of endurance and discernment (noting that discernment is also useful for racing, but endurance will affect normal riding time too) to your liking. If your stats aren't where you want them by day 63 or whatever, when care plans stop doing much of anything, adjust them after that with chocobucks training.
Note that there's a limit to the sum of your stats (rather like 60+ craft skills), so you can't just raise *all* the stats up high. You can have two max and two low, you can have one max, one high, one moderate, one low, you can have two high and two moderate, etc.
For more details, the best source of info on choco raising that I've found is FFXIclopedia's forums (specifically the choco raising/racing forum, which is in the crafting forums section). If anyone has another good source of info, I'd be happy to learn about it.
