I am brand new to the game, but am falling in love. I found an egg in the wild, and it brough up some questions.

1. I accidentally moved it to my stable. Can I move it back to my hatchery?
2. How do you even hatch eggs? I assume it has to do with nurturing, and i've been doing other's eggies all day.
3. What's the real difference between the Hatchery and the Stable? Do eggs hatch faster there?

Thank you so much! I love you! Be safe, and any advice is appreciated!

1+3. you cannot move things from stable to the hatchery. however, whether an eggling/hatchling is in either makes only one difference- that being, they need to be in the stables to be put up for public sale rather than private sale only like in the hatchery.
Regardless of where they are they can still be nurtured per usual, either on their pet page or via the nurturing grounds.

2. Time and nurtures both progress the growth of egglings and hatchlings.
We cover aging of sylesti here
but in short, egg+hatch age at a rate of one hour per hour, but a certain number of nurtures (5?6? for egglings, 4 for hatchlings (it's lower for hatchlings to balance out that they get +1 to their lowest stat per every 8 nurutres)) will jump them forward an extra hour

Thank you so much! This clears a lot up. You're wonderful!