I had trouble with this too. This reddit guide, and the resulting discussion, greatly helped me understand the mechanics. Im sure there are innumerable ways to use characters to be effective, so looking at some examples of how people are getting through it may be useful.

This Youtube video may also be useful in this capacity as an example of how to beat each type of shield.

From the wiki:

Abyss Mages creates an Elemental Barrier surrounding them, preventing anything from dealing damage to their HP, in order to destroy it quickly, you will need to use an element specifically that creates an elemental reaction.

Tips:

Some enemies like the Pyro slime is immune to Overload Elemental Reaction due to the elemental damage it deals, but Elemental Reactions happening on an Electro Slime deals devastating damage to them due to Overload dealing pyro elemental damage.

Freezing enemies when Freeze Elemental Reaction happens increases physical damage, dealing any other elemental damage towards the enemy would only waste the opportunity for a hard shattering physical damage bonus.

When Electro Charge Elemental Reaction happens, any enemies nearby that is Wet will be dealt Electro damage overtime, if enemies or any other characters is standing on a wet or flooded floor, there is a chance that you will get yourself electrocuted and you will be inflicted with electro damage until the electrocution on the water disappears.

Some enemies are not elemental base, but particular ones are immune to some or specific elements, if you don't have a specific element to counter this, your only way of dealing with them is using physical damage.

Every Character that uses a book, or an orb is an elementalist, they are more effective in dealing elemental status on enemies and inflict or create elemental reactions efficiently.

Pyro Abyss Mage

Elemental Weaknesses:

• Hydro
• Electro

Elemental Effects:

• Vaporize: Increased damage, element effect dissipates
• Overload: Explosion Occurs, dealing additional Pyro damage

Cryo Abyss Mage

Elemental Weaknesses:

• Pyro
• Hydro

Elemental Effects:

• Melt: Increased damage dealt
• Freeze: Freezes enemy, increased physical damage. Frozen enemies when shattered, deals bonus physical damage

Hydro Abyss Mage

Elemental Weaknesses:

• Cryo
• Electro

Elemental Effects:

• Freeze: Freezes enemy, increased physical damage. Frozen enemies when shattered, deals bonus physical damage
• Electro charged: Deals Electro damage overtime for a short duration.