I have right now teams of these elements, but they are not lasting long in the festival zone against bosses. I didn't do very much combat until now.

Water Wind Fire
Wind Fire Light

I bought some higher stat pets to make a new team with but now I am not sure what the best set of elements would be except something different than what I had before. I would like to not need as many potions as I do now.

These are the pets I plan to use for the new party:

https://www.sylestia.com/view/pets/?petid=4765912
https://www.sylestia.com/view/pets/?petid=4160553
https://www.sylestia.com/view/pets/?petid=4344388

i like water-light-air. personal opinion, but that's what i like. water-air-fire should be very good at doing damage, a tank and 2 cannon types. air is better for bosses b/c of the higher damage to a single target.

here's your trouble:
current party pets Mulliran, Runica, Flurian
first off they have unattuned prof points. agility works good for tanks, recommend int/str in the other 2.
also they're really bad armor prob isn't helping. yes, those are the right types- assasin, evoker, myrmidon- but those are really low levels (runica has a lv 30 bracelet) and they aren't quite as good with boss battles. with the higher difficulty in 60+ in the fest zone, you need exotic at least, but preferable vinethorn equipment. both are from tlg.

Personally, I've been using shadow/light/air since the lost grove was released and haven't looked back. It's a really good basic team setup and I can do the festival zone easily. These three are my party if you want to take a look for how I've allocated proficiency points. They are pretty maxed out with gear and stuff as well as being max stats, which probably helps:
shadow, light, air

If you wanted to try out this team build, I'd recommend your nephini as light, griffi as air, and ny'vene as shadow, just based on their stats.

our own team is Earth/light/air- not max stats, light pet is set up differently from Wavypavy's (air is, because Air is just... best that way, 3str/2dex/all storm caller)

Really a good team depends on your own style- you just have to find a team setup that really clicks for you, meshes with your battle/strategy.

(we also run lv50 battle board teams seen here (save for the Luffox team's light tank Spider Sprite who I have Only Just Noticed has low level bracers no wonder he was getting hurt more than he should earlier) and here (max lv50 light/fire/fire nytes))

Thank you all for the advice! I see that against bosses air is much better than fire so I will be using that for the main damage.

I see one of you has health and agility for the light user while the other has intelligence and dexterity - could you explain your reasons for this difference?

Agi lets a light pet get the heal off faster- higher chance of going before the opponent moves-, dex increases the chance of Beam and Fairy Light getting critical hits, and thus healing the team for more.

Our Edge of Atrocity has some lightbringer equips rather than full pyromancer specifically for that reason- just a bit more speed, rather than straight-up dex, so she has a very nice chance to hit a critical, but she's fast enough to go before many enemies.

Might I add that having a tank attuned to Water is decent because it can, at higher levels, grant Ice Barrier and Tidal Surge to help the team.

Ice Barrier helps to protect a vulnerable pet while Tidal Surge increases the damage the entire party deals.

I am thinking of starting a secondary party and while I like Earth, it doesn't really help out any other pet in the party plus I don't like "Taunted" because of your tank is already under fire and dying, attracting even more attacks by using Leeching Strike or Mocking Blow is suicidal.

The problem is that you need a battler to do the heavy fighting but having another pet that can help out is good too and often that task falls on either the tank or support from a Fire user.

HOWEVER...the party needs to be healed overall so a Light pet is almost a must.

Right now my party consists of a Water tank (Butternut Squash), an Air battler (Regal Cutlass) and a Light pet (Mangle)

I like to get a battle over and one with before my adversaries can do much damage so my only choice, really, is to have either a Shadow or Water as a tank, Air as a fighter and either a Fire Bender or a Light Healer.

Fortunately I bought a level 66 pet with pretty good stats (wish it could be max but..) so maybe if I make it a Earth tank then it can help out in fights too....