I found an egg when exploring. What do I do with it? How can I hatch it?

it goes to your hatchery, and will hatch around 3 days

Welcome to Sylestia. ^^

Just like Snowcrystal said. The egg is in your hatchery and usually takes around 3 days to hatch.
Once it hatches it will become a hatchling (baby pet)
Additionally once a day you can nurture (click) your egg or hatchling in the hatchery :)

For the eggs

If it's cold - warm it
If it's warm - cool it
If it's restless - turn it
If it's dull - polish it

You can also nurture other player's eggs and hatchlings :)
And other players can nurture yours. It helps the egg hatch faster ^^

For the hatchlings

If it's weak - work out
If it's derpy - read a book
If it's restless - play with it
If it's slow - race with it
If it's hungry - give it food

I hope this helps! Have fun here on Sylestia! ^^

Thank you both so much!!!