Im new. And Im trying to breed some sylvesties, how long does it take?

You say catch but then are asking about breeding time so i am assuming you mean the whole growth process instead of just the catching totorial ... ? Please tell me if i am wrong here.

How long until a pet can breed for the first time depend of the maturity of said pet :

In the case of generated pets (eather from essences or your starter) it take 18 days for that pet to become adult.

For hatched pet, without nuriture boost or magical pie, it take 26 days.

For caugh adolescent/hatcheling it depend entierly of their maturity at the time you do get them.

You can see the maturity points of a pet only after it was tested (unless generated), this will allow you to see how far along they will be from becoming adult. 4 nurriture click add 1 point but they also increass naturaly every hour, from creation of an egg you nead 648 points for the pet to become adult.

Sorry I didnt explain it right! Im saying that Im catching pets for breeding. How long does it take for the pet to be catched?

Just the time of the battle there isnt a wait system for those beyond the turn based fight, all you nead is a few traps and your team, it isnt a garentee thing tho and some wild pets can flee after you send a trap.

So do I throw a trap, take it back, then throw it again?

you dont get the trap back once you throw it, eather you get the pet, the flee or it stay and you can throw another trap at it. It's a similar catching system to the one in pokemon, you do nead to select the target when you click on the trap tho as it does work even if there are multiples wild pets on your screen.

How do I know that the pet is out of the trap?

If it escape it will be facing you in the battle next turn, if it run away or if you caugh it it wont be there. To confirm that you caugh it you can check the log beneth the exploration window or check your stable for your new pet.

Thank you!

Just an FYI, you can certainly catch pets but I would highly discourage it. Traps are really expensive when you first start out and pets almost always run away. It is way cheaper to go to the advanced search and purchase pets there.