Apparently pets are kinda like guest characters that appear only at certain times throught the day when you press the C button.
Here is how to get a family pet.
First off, you'll need a Mametchi, Memetchi, Kuchipatchi or Furawatchi/Violetchi. Once you have gotten one of these characters, raise bonding percentage to 100% using bonding items, connection games, training or whatever it takes!
Next step, get a friend with the required character or keep on checking the dating channel until you get the required character to mate with.
If you have a Mametchi, the required character is Chantotchi.
If you have a Memetchi, the required character is Mumutchi.
If you have a Kuchipatchi, the required character is Yonepatchi.
If you have a Furawatchi/Violetchi, the required character is Sukatchi.
If you are mating via connection, you must receive the Chantotchi, Mumutchi, Yonepatchi or Sukatchi. Do not send your character or it will not work.
If you are mating via the dating channel and you've already checked the dating channel three times to no avail, try setting the time to either 11.59pm or 6.59am and wait a minute... I'm not sure what you're meant to set the time to, so try both! Anyway once you do so you should be able to use the dating channel three more times. Keep repeating if you still don't get the required character.
Final step, you should get two babies. Check the status screens, if a special family bar appears, you know it's worked! The two babies will evolve straight into the two special characters after an hour. The characters could be any of the sets below depending on what family you have. I'm also going to put the names of the pets below as well.
Mametchi and Chamametchi (with the pet Bagubagutchi)
Memetchi and Imotchi (with the pet Chitchi)
Kuchipatchi and Chibipatchi (with the pet On Sen Moguratchi)
Furawatchi and Kizatchi (with the pet Hanatarezoutchi)
The pet will appear at certain periods throughout the day.