Hi! I just joined today and I have a question. While I was doing the tutorial, I got an egg, that after the tutorial I found in my hatchery. What do I do with this/When will it become a Sylesti?

Eggs will hatch after three days. Some can hatch faster if you ask people to click them. Hope this helps, if you have questions, feel free to ask!

Further clarification:
Eggs have a 72 hr. timer that can be seen at the top of their Genetics section as x/72.

Eggs found while exploring tend to already be at least somewhat progressed, so only bred eggs will start at 0/72.

You can nurture egglings/hatchlings once per day, and for every 5 successful nurtures from people egglings will have their timer boosted by an hour while hatchlings will get +1 to their lowest stat (ties being resolved in order of HP(divided by 10?)>Str>Int>Dex>Agi)

Once the timer hits 72/72, there'll be a little symbol on the top of the eggling's slot in the tab, and when you click them you'll be taken to their page and get a popup that gives the options of Help it hatch (+20 HP), let it hatch on its own (+1 to one of Str, Int, Dex, and Agi randomly), or use a Philter (select from the ones you have via the dropdown)
If you do not interact with and egg and just ignore it, it'll eventually just hatch on its own with no bonuses.