How do you get an egg to hatch? And how do you get an adolescent pet to adult level?

They hatch and grow by age
they hatch at 3 days
they grow to adolescents 6 days later - 9 days
the grow to adults 18 days later - 27 days

Time is the way a pet ages. However, nurtures for an egg can speed up the process. Every five nurture clicks is an hour off the time before it hatches. Nurtures on a hatchling increase stats and page views also give it maturity at 1 maturity point for every 5 views. Aside from this, however, it is just a matter of time as has already been outlined.