Ive been taking my pet to the gym but what other ways can the stats increase without having to wait so long?

There are stat boosting items out there, but I feel it would be best to wait out for when you training times get lengthy (mine are at 6 and a half hours for example). There is also the CSC feature of autotraining, which will give your pet 24 hours worth of courses so it could be as if you have been training for 24 hours. Also, the EXP you get from battle quests can help, especially when your pet can take on bigger opponents.

Do I have to go somewhere with exp points?

You go to the "spend EXP" page of the battle coliseum. You can use it any time you want, whether it'd be now or when your pet gets bigger and their courses start taking longer.

yeah, i would just do battle quests for now. When you want to get stat boosting items I would do shinwa's quests. You can do them once a day and though she doesn't give out stat boosting items all the time, she does give them out pretty often imo which is helpful since they're kind of expensive