I keep losing to Cereth in fungus grove. It's costing me so much money every time to keep reviving my Sylestie. How do I beat them? Are there certain tips/tricks or do I just have to keep trying and dying?

mmm.
1. Pets will revive without healing given time so long as they aren't in battle.

2.
Your issue probably mainly lies in your equipment
At the very least, you'll want to swap Kelp and Ocean's Relics, because at the moment Ocean (Water element) has a Magical damage-boosting relic and Kelp (Fire element) has a Physical damage-boosting relic, which is the reverse of what they should have.

Also try to get them either some higher level or higher rarity (or both) equips, and fill all four equipment slots.
At the moment they're both running Lv0 or lv5 equips and only Ocean's armor isn't Uncommon rarity.
Lv15 is the highest they can equip right now (if nothing else, if you send them on some Medium difficulty missions (the Str. one for Ocean or the Int. one for Kelp or Peppermint) you should be able to get some lv10 rare equips (also can give gold and healing items, so))

3.
Battle strategy, go for one guard, then the other, then aim for Cereth. Make use of Mend ability and Finishing Blow.

thank you!