I trained up my pet, but I'm not sure how to spend the points to up his stats. Can someone tell me how?
Do you mean you earned exp? Because that can be spent here: https://subeta.net/games/battle/exp.php
In addition to what Stephen said, if you've been battling, you should have some tokens which you can spend in the Battle Shop. There is some stuff in there that can raise (and lower) a pet's stats.
No, uh -- I went to the training field? And I trained my pet, and it said I would get "level points"? I just wanted to turn up his points so I could put him on a job.
Ah okay. If you want to get a job, each job has a certain level you have to reach, a certain amount of strength you have to have, and a certain intelligence level you need.
You gain higher levels and more strength through the Training Center, and you gain a higher intelligence by reading the pet books.
These aren't "points" in the sense of currency. They are just the amount your pet has.
For example, Kyoka Jiro has Level 1, Strength of 10, and Intelligence 0. So she could only get a job that has an entry requirement of 1 or less levels, 10 or less strength, and no intelligence.
My pet [user not found] has Level 128, Strength of 324, and Intelligence 347. So she can get a job whose minimum requirements are anything less than those numbers.
Keep in mind that if you want your pet to get promotions in their job (which mean more sp earned), then you will need to keep meeting the new criteria for each promotion.
This thread is very helpful for finding out what level, strength, and intelligence you need to meet for each job promotion, so you can plan ahead: Job Agency Guide and Discussion