A lot of the responses have unclear answers or incorrect information. I will explain to you how to get Mametchi and the Mame Family in the easiest way possible:
How to get Mametchi:
The easiest way to get Mametchi is to keep bonding high (80 - 100%) and always answer the training calls with the Smart option. The Smart option is the leftmost training option when your Tamagotchi calls for training (when the 6th icon is highlighted). If the bonding range is any lower than 80%, you will not get a Mametchi. If you do not pick the Smart option when answering training calls, you will not get a Mametchi. Please remember that you can get Mametchi from any male teenager.
How to get the Mame Family:
To get the Mame Family, you will need to be a Mametchi. Follow the steps I wrote above to get him. Once a Mametchi and the Dating Show becomes available, select your Mametchi to marry and visit the Dating Show until your Mametchi is introduced to Chantotchi (you can only visit the Dating Show 3 times a day, but you can reset the clock to visit the Dating Show more often). Once your Mametchi has married the Chantotchi, he will have two Mimifuwatchi children and leave them. After an hour at baby stage, the children will skip all other growth stages and evolve into Mametchi and Chamametchi. The parents will be Papamametchi and Mamametchi, and you will have the Mame Family. During family animations, you will also see Bagubagutchi, the family pet.