I gotta confess, I'm stuck on the tutorial. But I'm chalking that up to my own stubbornness and general lack of patience, yikes. EDIT: I completed it :D
I was wondering if there were any tips I should have before I dive headfirst? So is battling the major premise of this game? Like Pokemon? EDIT: How do I add new slots to my party? I'd like more than 1 pet battling at a time.
I'm a player on Flight Rising, and I see some things are similar (thankfully, I'm not going to be competely unprepared.)
But, uh, yep. I'm excited to start.

You can get a second slot by buying it for 20k or so by clicking on esior's haven and then sylestia plaza. However, as a guide in general, I recommend the following two posts:
https://www.sylestia.com/forums/?thread=64486
https://www.sylestia.com/forums/?thread=45988

Additional help can be found by asking players or by reading guides.

I would strongly recommend getting the extra pet slots as soon as possible, as well as an extra pet to fill the third slot. That will make battling so much easier! Second slot is available from Esior's Haven, and the third can be gotten from Sanctuary of Saerielle, if I remember correctly. I couldn't tell you how much they are worth, though.
You don't have to spend gold on healing your party pets up every time they all get knocked out unless you want to keep battling immediately; they'll heal on their own over time.
Participating in the site events where waves of enemies are invading a zone is always a good idea; you can get event satchels with items, gear, and gold in them just for defeating a minimum amount of enemies! The enemies are scaled to the level your pets are at, so it's just like progressing through a zone for your own purposes.
The Festivals are bigger events, and also worth participating in. The next one is the Fall Festival, and is happening very soon! The Festivals give the opportunity to catch wild seasonal fabled pets (Ryori for Fall, Griffi for Winter, Aeridini for Spring, and Lighira for Summer) as well as wild Themed pets for all species. Both of these types of pets are rare, but it is very possible to encounter them. This can be a nice way to add to your stables! You should definitely try to catch every themed pet you come across, regardless of if it has any visible traits or if it is a species you like: there will be someone willing to buy it! And I personally always go for any fabled pet with a visible trait.
Diamonds are the premium currency, but they can be bought from other users with gold. Just keep in mind that the prices tend to go up before and during festivals.
Anyway, welcome to Sylestia! I hope you enjoy the site!

I wouldn't say battling is the main premise of Sylestia in the way it is in Pokemon. There's no one thing that you could say is the main premise, except generally collecting pets. I personally find breeding the most enjoyable aspect of the game.
Something lots of people struggle with when they're new (myself included) is getting gold. I find that the best way to do this is by sending pets who aren't in your party on missions at the town barracks.

Thank you all! This is all genuinely helpful stuff, and I can't thank you enough for sharing.

Welcome! If there's anything else you want to know, feel free to PM me; I can't guarantee I'll be able to give a good answer, but I'll definitely try!