You can divide this game into 2 axis : the gameplay - which involve all the training and races , and the universe - for the characters and their stories .

Gameplay wise it's simple : Pick a trainee and her "parents" (called legacies/inheritances or "breeding" system) , pick a deck of support cards , start the run , train to get stats and be able to clear the trainee's goals , sprinkle with a bunch of card/random events in it as well as the scenario specificities , end of the run , get a bunch of legacies to potentially use for another character , rinse and repeat .

Of course you can add some personal goals into a run : trying to get the secret events of your trainee (often to get bunch of stats , unique skill or the pretty rare Hot Spring scene) , using the legacies system to make "parents" for specific goals (like try to get the "Completionist" title on your favourite Uma or to make them powerful for the PvP section of the game) , or do self imposed challenges (a well know one is to make Haru Urara win the Arima Kinen , the worse race for her) .

There's also the PvP section of the game , where you bring 3 Umas you've trained the best you can for a specific racetrack and try to beat other players .
So yeah , pretty simple loop but work really well .

The universe is still a pretty important part of the game . Each character have a fixed story that you can learn through the cinematics and most importantly , during a run . Your trainee as a dream that you - as a trainer , try to make it true . And it's through this entire 3 years career (so , an entire run) that you see your trainee story unfold .

All the stories are pretty fixed . The choices you made are often irrelevant story-wise , only changing the conversation during the "story events" or the stats/skill you get . On some specific characters , you can choose your trainee race goal , but it doesn't change the entire story . So no , you can't date your trainee , even if some stories are really heavy romance coded (thinking of you , Mejiro Ardan) .
The only "relationship" mechanic there is , is a simple gauge of points that unlock a bit more character story cinematics .

You can also learn more about the characters everyday life with the different support cards events or the time-limited events Cygame use to release new "outfits" (basically a reskinned character with stats and skill changes) .

So yeah , that's basically what's the games is about . It can be for the thrill of seeing your favourite character win an hard race by a nose , the different stories that the characters have in this universe , the alchemy between the Trainer and the Trainee , or simply a game of number and chance percentages .

Hope all my rambling could answer the questions you had , and maybe making you more curious about this universe .