I'm going to analyze all the major runs and see which armor and defenses are optimal for them. At the end, just to satisfy my cheap side, I'm going to list everything that can be obtained as cheaply as possible.
Gloaming Wildwoods
Dominant families: beast/construct
Damage: normal
Weaknesses: pierce, elemental, shadow (monster cages)
Best shield: Ancient Plate
Best armor set: Ancient Plate
Royal Jelly Palace
Dominant families: slimes/constructs
Damage: normal, pierce, elemental
Weaknesses: shadow, elemental
Best shield: Ironmight Plate
Best armor set: Deadly Virulisk
Ironclaw Munitions Factory
Dominant families: constructs/gremlins
Damage: normal, elemental, shock
Weaknesses: elemental, shadow
Best shield: Grey Owlite Shield
Best armor set: Mercurial Demo
Firestorm Citadel
Dominant families: undead, fiend
Damage: normal, shadow, elemental, fire
Weaknesses: elemental, pierce
Best shield: Volcanic Plate
Best armor set: Volcanic Plate
Shadow Lair Gloaming Wildwoods
Dominant families: undead, beast
Damage: normal, shadow, poison
Weaknesss: elemental, pierce
Best shield: Dread Skelly Shield
Best armor set: Valkyrie
Shadow Lair Royal Jelly Palace
Dominant families: slime
Damage: normal, pierce, freeze
Weakness: shadow, fire
Best shield: Heater
Best armor set: Ice Queen
Shadow Lair Ironclaw Munitions Factory
Dominant families: construct, gremlin
Damage: normal, elemental, fire, shock
Weakness: elemental, shadow
Best shield: Grey Owlite Shield
Best armor set: Grey Feather
Shadow Lair Firestorm Citadel
Dominant families: undead, fiend
Weakness: elemental, pierce, freeze
Best shield: Crest of Almire
Best armor set: Deadshot
What's the most 5* stuff you can get spending as little crowns and energy as possible (i.e., cheapest)?
Note that the Shadow Lair armors require no recipes, and hence no cost there. But they won't benefit you if your Shadow Lair runs aren't profitable.
Azure Guardian set
Mercurial set
Royal Jelly set
Ice Queen set
Ancient Plate set
Crest of Almire
Ancient Plate Shield
Heater Shield
Royal Jelly Shield
Barbarous Thorn Shield
Aegis
Fang of Vog
Voltedge
Triglav
Blitz Needle
Plague Needle
Argent Peacemaker
Sentenza
Biohazard
Neutralizer
Polaris
Supernova
Nitronome
Electron Vortex
Voltaic Tempest
Dark Briar Barrage
There's no such thing as cheapest, it's just either general (anything you can buy the recipes from hall of heroes), event (kat) or boss items equipments. Unfortunately, the armor set always lean toward offensive bonus, as it is actually makes you safer. With damage bonus, you can knock monster easier to cancel their attack charge. On shadow lair, although you get the correct equipments, the defense is still not easy, I think what matter the most is how you react against monster attack, not the defense itself. Grey owlite shield could break within two rockets hits on shadow lair Roarmulus twins.
I would say just stick with offensive bonus armor set, and matching shield if the area gives you hard time. Otherwise when you feel the areas are easy, just stick with barbarous thorn shield or swiftstrike buckler.