How can I raise my pets strength and level without using the training center?

There are stat boosting items. You'll need something that will raise the stats you want.

https://www.subetalodge.org/list_all/search/Stat+booster/mode/notes

If you check the individual page for each item on SubetaLodge, it will tell you which stats they boost.

Note that some boost more than one stat, if there are stats you don't want to increase.

Also, some give a boost of something like "0-1". You could get ZERO stat boost upon using these items. If you don't want to risk that, make sure the item guarantees at least 1 point in the stat you want.

Battle quests are also a good way to increase your pet's status. You earn XP from them, and you can convert them into status here. You can also get XP by battling the opponents in the Battle Coliseum.

There are also a few other ways not mentioned:

1. buy scratchcards - one of the winning options is stat gains. The stats go to your active pet.
2. get a job for your pet(s) at the Job center. Every once in a while when you collect your pay the pet will get a stat gain.
3. roam around the site a lot to generate random events, one of them gives your pets stat gains.
4. another random event is Oracle quests and the reward for those is always stats (plus some other stuff)
5. If you do battle or battle quests you will sometimes get points to use in the battle shop - in there you can buy stat leaches, that way if your pet gains stats you don't want, you can delete those.