Your tama does have to be an adult to adopt a pet. You'll find them when you visit the park and you'll need to play with the same one a few times before you get the option of adopting it.
Yes you have to be an adult, you have to go to the park and there is different pets and different times of day when you have played with the same pet 3 times as an adult it will ask if you want to take it home with you.