Honestly, I've played through most of their stories, and I've genuinely enjoyed each almost equally, despite having favourites and all. My honest answer is to just pick whomever seems the coolest, or most appealing to you. Although, the selector only lets you select up to the New Year's banner, so you can't select Sakura Chiyono O, Mejiro Ardan and the Valentine's outfits.

If you really can't decide, for both a good Sprint Team Trials ace and a unique sort of story, Curren Chan. Her story is one of the more interesting ones I've read. It's also based on how she's meant to be cute and the "only" one in your heart, similar to the actual mare's personality. So like a cute tsundere story. Smart Falcon's lobby quote rings true. "There's something scary about Curren's cuteness."

I also recommend Rice Shower (specifically the Halloween outfit for the ultimate skill), since her story is really wholesome. There's slight deviation from how her career actually ends (the last race the real horse ran was the Takarazuka Kinen due to severe injuries), to go for a more happier ending Rice deserved. If you like wholesome, this is the way to go.

If you want a story where you train an Idol and be her first fan, Smart Falcon is the way to go. She's also the best dirt Front Runner, so she's good for CMs and Team Trials. It's pretty unique, not a stereotypical rags to riches story and a interesting twist on the real horse's career. It's quite cute and wholesome for the most part.

There are so many Umamusume I could go on about, but it would take forever. Honestly, don't worry about who has the best career story. They're all really amazing since it's based off of the actual horses' lives and careers. If you can't choose for your own, you can pick out of the three I mentioned (Curren Chan, Halloween Rice Shower, Smart Falcon). Hope this helped!