I found an egg while exploring, clicked on it in my hatchery and it gave me a prompt and some options. Is there a certain way to take care of it? How do I know when it's time to hatch?

When it gives you a prompt there is a related answer for every one, for example if the eggling is too warm you want to click the cool eggling button. You can monitor the growth of you eggling by a bar towards the bottom of its profile page. You can only nurture once in a 24 hour period I believe.

Here from the newbie tips thread:
Pet Aging and Maturity
Maturity is measured in hours and comes in four categories: Eggling, Hatchling, Adolescent, and Adult.

It takes roughly 72 hours (3 Days) for a newborn Eggling to become a Hatchling.
It then takes roughly an additional 144 hours (6 Days) for a Hatchling to become an Adolescent.
It then takes an additional 18 Days for an Adolescent to become an Adult.

Profile Views and Nurture Clicks can speed up the growth of Egglings and Hatchlings. Egglings and Hatchlings cannot take part in exploring of any kind and must stay in your Hatchery until they are an Adolescent.

When an Eggling reaches 72 Incubation (The Green Bar on their Profile Page), it will be Ready to Hatch. You can either choose to Help It Hatch, Let it Hatch, or use a Philter. All three choices will permanently increase its stats. Which stats and how much depend on your choice. Also, if you wait too long, the Eggling will just hatch on its own and you will miss this opportunity.

alright, thank you!