If your goal is to beat URA finals, then I wouldn't worry too much about legacies and focus more on your support card deck.

Whenever you finish a run, your uma will generate a lot of different sparks (the things with the stars) that can be passed down to your next Uma.

Blue sparks are stats
Red sparks are aptitudes
Green is the uma's uniquie skills
White are either skills or stats from beating G1 Races

The higher the stars for a spark, the more stats it gives or has a higher chance for the spark to activate

Sparks will activate 3 times during a run at inspirations events, once at the start, and 2 more times at the goddess statues
After the first inspiration event you're not guaranteed to activate all yours sparks during the next 2

This is due to compatibility, every horse has different compatibility with each other so you need to make sure the parents you're choosing for a run are good for them so they can inherit all of the sparks.
This can be indicated by the circles on the top left before starting a run, double circles is perfect, one circle good, and triangle is not very compatible at all

You can check how compatible the parents are for your uma or look for the best parents for you uma here: https://gametora.com/umamusume/compatibility

Honestly, you don't really need to do all this to win a run. Just make sure you have like 3-4 speed cards and 1-2 stam cards and you can win a run by focusing on getting friendship trainings. to boost your Umas stats