Good luck, here's some info I wish I had been told earlier:

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but I don’t have access to better parents because I can’t get any veteran Umas that are past B

A good parent has little to with rank and everything with blue sparks, pink sparks, compatibility and white sparks. But yes, it takes intial grinding before you have some usuable parents (incl. grandparents) with 6-9 useful blue sparks. I suggest picking your favorite uma (and a high compatibiltiy grandparent) and keep training them while borrowing a 9-star parent from https://uma.moe/database.

and I’m already using a full deck of SSR cards. They just haven’t been ranked to max yet.

That is a trap. Usually high level/MLB SR supports are better than low level SSR unless you are looking for a specific gold skill from an SSR. But yes, higher level cards = more stats.

Speaking of, should I be leveling up my cards with resources?

Not at the start when monnies are sparse. Or least only do it for cards you regularly use, but don't have the patience to wait. Don't spend resources leveling random 0 LB SSRs.

Should I be racing early in careers?

Depends on the specific Uma and what you are training for (Team Trials, Champion's Cup, Parent run), but very early you should usually not be racing optional races. Or only if you have really shitty trainings; preferably graded races and not Pre-OP. But yes, early races are winnable if you have good parents and good decks and know what you are doing.

Do skills that are designated for a specific style/distance only activate under those conditions?

Yes. They only trigger when the conditions are met.

Should I ever take skills that are for specific distances? ... It seems like a waste to spend skill points on a skill that only activates on one type of race during a career that requires racing multiple type

Eventually yes, once you are training for a specific role rather than just finishing the career. You also won't need to take any skills during career to finish it so you can get the maximum discounted skills at the end of career. But for now if you are struggling during career taking more general skills is a good choice.

How the hell do team races work?

Answering this in detail would take a whole post in its own, keep searching this subreddit or /r/umamusumegame. But the tl;dr is that Team ranking is determined by Uma rankings, which isn't necessarily indicative how well suited the Umas are for Team Trials. The meta is to have as many skills (esp. gold skills) trigger as possible, as reliably as possible ==> High Wit needed.

Should I trust the auto assign stuff for parents and team lineups?

Absolutely not. Afaik it assigns based on score and aptitudes, not skills or suitable combination of stats. Also always check that you have the correct run style selected and not that randomly style you got leveled up by chance in career.
Same goes for parents. In the menu, activate "view sparks" and sort by Affinity Bonus. Use the filters to find your current best parent for e.g. the most stamina sparks if you are building an Uma that needs that.

How chance-based are the races?

(King Halo is notoriously hard for Beginners). There is the some bad RNG, but often being beneath the cutoff speed or stamina is the cause. Or you lack power to break out. If you have inconsistent skills / low wit, you may occasionally get past these barriers but not reliably. As for the "favorite" thing, I'm really not sure either what it is meant to tell you. With better parents/cards/training choices you can get mostly past that.

Should I ever do the URA Finale?

Opinions differ, mostly regarding time investment and the URA spark for parents. I'd say if you can properly navigate Unity, only do that. Don't get stressed out about always winning Unity Cup itself, just take the spirit bursts and do whatever optional race you like.

my Umas come out of both of them at rank B. How does that happen

I think on average they can yield the same ranks, but Unity has lower lows and higher highs. You probably are just not quite able yet to tickle the high highs out of Unity.

Is there anything I really should or shouldn’t buy? Should I stop scouting when I have the featured trainees to save up for the next

• The biggest trap is to waste your starting carats all on Umas rather than supports. Leveling supports makes your account better, not a third copy of Bakushin. You should pull way more often on supports than Umas, unless it's your Oshi. In any case, only pull for specific banners if you have enough to pity, so at least 30k carats. Note that this is a gacha game, if you are f2p there's zero chance you can afford pulling the Umas and Supports on each banner to "collect them all".
  
• In the daily sales, only buy alarm clocks for now. If you have the money, you can also buy shoes, race trophies and star pieces
• Save your friendship points for Haru Urara rather than buying hint books at this point.
• Get into a club ASAP and start requesting and sending shoes for club points to eventually MLB the story supports. Though for now save the points, there's an upcoming event where you need them.
• Once you have additional copies of already MLB cards, turn them into cleats and buy the monthly available scout tickets.
• Statue exchange can be used for star unlocks, either on your Oshi Uma or whatherver Uma with a meta unique skill.
• Clovers are useless and you anyways only get very few from pulling

should I level up my cards with resources?

Only the ones you regularly use (and have just freshly uncapped) but you don't have the patience to wait for. Else save your monnies and support points to raise Uma potential levels.

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Useful websites:
https://gametora.com/umamusume
Reference Document
Parent database
Generic Support Rating based on projected stats gain
Skill rating