Newbie here, I've kind of accidentally had my starter pet pass out or die. I'm not really sure, his health dropped to 0 in a battle. What do I do?

The pet, luckily, hasn't died, pets on Sylestia do not die. They only end up unconscious when defeated in battle.

This is a quote from another user(Thanks so much to them for having previously explained this), that explains the way to revive your pets:

Link: https://www.sylestia.com/forums/?thread=89438&page=1#2 Author: Scathreoite Time Posted: 5/25/2020 at 3:40 AM this should be in the Q&A forum Anyway, revival options > in the stables, there's a button next to the 'feed pets' button labled 'revive pets' > pets that are not in need of feeding will revive on their own 30 minutes after being knocked out > Revival draughts (obtainable from Esior and Saerielle town markets, and a possible drop from mission satchels > the healers (located in the plazas of Esior and Saerielle, one in each section of the fest zone map (one near the entrance and one next to the house). Can also be occasionally randomly found in the Dungeons.) Generally speaking it is not recommended to use the healers because uh. They're scammers. They charge way more gold than you reasonably should pay.

The best option would likely be, since you're newer and likely don't have much gold, and may not have revival draughts yet, to let your pets revive on their own. After the pet revives, they'll regain health every 5 minutes if I remember correctly.