I was wondering if there were any other ways of capturing pets aside from setting traps?

Also, when is the best time to set a trap? Does it make any difference if I weaken the sylesti first then set the trap?

Sometimes you can find eggs and baby hatchlings while exploring, you don't need nets to capture those but Adolescent and Adult wild Sylesti always need traps to be captured.

I don't believe weakening it has a huge effect on the catch rate of wild pets; it does increase your chances, but not by very much. The best advice that I could give is to try to have the wild Sylesti down to about half life, then start using moves like "Mend" and/or "Block" to keep from hurting it and accidentally killing the pet you want to catch. Also, have a good amount of traps in your inventory as often as you can; you wouldn't want to find a neat pet that you want, but not enough traps to capture it with.

Thank you Edibbs! :) I guess I need to play for a bit longer before I can have a good enough amount of traps... I'm too poor to buy more than one at a time, and my sylesti don't have Mend or Block moves yet.