I have two questions to ask. The questions are how long does it take for one of the animals to grow up and what are the ways of obtaining an animal?

I have the same question too.

Continue the storyline and Jinxie will give you a scripted quest to catch a pet from the wild (which you don't have to buy traps for). Aside from capturing pets, you can breed them, find eggs while exploring, find essences while playing (you can generate pets with these), win them in holiday events or buy them from other players.

Edit: Pulled from Xavion's guide, going by maturity found in the the hex code:
At 0 a pet has just been made/born.
At 72 hours, it's ready to hatch.
At 216 a pet is officially an adolescent.
At 648 hours it's officially an adult, and ready to breed.

Thanks!

Thanks.

I was also wondering how to buy pets? I looked in the trade section but I couldn't find where.

you have two main choice for that :

-Advenced search (https://www.sylestia.com/search/) who can be helpful if you look for something in particular, the best way to found a cheep pet with visible (if you don't look the colors too much).

- Pet auction, you can found easily some pretty pet but they can be more expensive.

An other way to get pet is when you participle to a contest but the chance are uncertain to win the contest. There is some giveaway sometime so keep an eye open to those ^-^

Okay, thanks. I will see if I can buy one. :)