The tamago date place opens up when your tamagotchi turns 6 years old. This is where you can find your tamagotchi a mate. The character you chose to marry will move in with you, and after a few days (1-2 I believe) it will then produce a egg which will hatch into a baby. Your adult characters will then leave your baby (48 hours after the baby is born) and you then have your next generation.
talking about married,yeah my mametchi and lovelitchi get married yesterday,for my tamagotchi it takes 4years adult and then you can get married after that having baby. and the father or mother will leave after 40 second from the baby born