The Bobcat is available for you to pet and cuddle (if you must) at random times on the path/area leading to the Town Center and the forest. Most of the time you can find the Bobcat from 6pm – 9pm.
The Bobcat can become your pet if you are able to pick it up three times. Once that is done, a memory / achievement will be triggered.
Another animal which you can have as a pet is your Dog. Unlike other Harvest Moon game versions wherein someone may give you a Dog as a gift, this version of the game will allow you to purchase a Dog. To get a Dog, you may go to Hunter and pay up 8000 to have a Dog in your farm.
To improve your relationship with your Dog, make sure to pick him up every day.