I just started today, and I was wondering how to get more pets, is the only way is by capturing them? Also, how does the aging process work? I found an egg while exploring and was wondering if it's by days, sorta like Flight Rising, or is it just a randomly determined time.

You can capture pets, breed pets (only adults can breed), obtain pets from other users, or generate them via the generator using pet essences.

Egg hatching can be sped up by people interacting with them, but I think they generally need about three days before they can hatch if I remember the rules correctly (You'll see a bar that says #/72 on the egg's page.) I think there's also an item that speeds it up but meh three days isn't that bad.

The best way to find pets for sale is to use the advanced search and then select search pets. It's under the search button on the left in blue writing. I know when I started I didn't know about it until someone in the forums mentioned it. There you can also search for pets to breed with.