How do my adolescents grow up to be adults? I'm having a difficult time figuring this out. Thanks.

It's all a matter of time :)

Eggs hatch at 72 hours.
Hatchlings turn into Adolescents at 216 hours.
Adolescents become Adults at 648 hours.

So there's 432 hours (648 - 216) between when a pet becomes an Adolescent and when it will become an Adult.

If you get pets genetically tested, then you can check their image URL to determine how many hours old they are. Example:
https://www.sylestia.com/image_builder_pet.php?species=ny%27vene&breed=sylesti&gender=female&gd1=DD&gd2=NA&gd3=ND&md1=NF&md2=NC&md3=BC&ec=f9f822&c1=18480f&c2=edf3a2&c3=0e5e0f&gd1c=83af49&gd2c=0e2606&gd3c=fafa73&md1c=18671b&md2c=0c6812&md3c=fbf76e&maturity=275&costume=0&owned=1&size=250

One final note, while the site has downtime, the hours stop running so that's why you'll sometimes see a disconnect between the number of days old the pet is and its maturity status.

Thank you very much. The info is extremely helpful.