Hello! I am relatively new to the game, and I am wondering how you get essences! I heard that you can get them from nurturing egglings, or by buying them from the diamond shop... But I am curious if there are any other ways! Thank you so much! Have a great day! :)

Sure, there are. For one, you can buy them for differing amounts of gold in the Trade Broker, but that isn't really a very affordable option, probably, for newbies. In the staff mini game in Jinxie's Jamboree, it is possible to rarely receive a Sylestie essence, and they are sometimes given as rewards following a successfully completed mission. In addition, one may obtain them as prizes from official events, and I believe I may have heard you can get essences from battle, though I'm unsure if that's true.

All of these methods only produce essences rarely, however, excepting purchasing them from the Diamond Shop and the Trade Broker.

I get essences from nurturing hatchery eggs/pets
(rare)

Yep, you can also get them from battles. As with nurturing, it's whatever species you were interacting with, and a rare chance.