Augusta Overview
Augusta, in my opinion, is a great unit for f2p players as she dont really need any limited support characters to showcase her full potential (unlike cartethyia for examplle), and has built in shield.

Her gameplay is using her abilities to gain prowess points, then convert these points into majesty points for an enhanced ultimate that doesnt cost Energy.

Skill Priority

• Resonance Liberation: ★★★★★
  
• Forte Circuit: ★★★★
  
• Normal Attack: ★★★
  
• Resonance Skill: ★★
  
• Intro Skill: ★★

Best Weapons

• Thunderflare Dominion (5★): 588 ATK, 24.3% Crit Rate. +12% ATK, +20% Heavy Attack DMG (15s) on Intro/Resonance Skill, DEF ignore on Shield (7.2%, 7s, 5 stacks).
  
• Verdant Summit (5★): 588 ATK, 48.6% CRIT DMG. +12% Resonance DMG, +24% Heavy Attack DMG (14s, 2 stacks).
  
• Aureate Zenith (4★) (new Pass weapon): 413 ATK, 40.5% Crit DMG. +23% ATK, +34.5% Heavy Attack DMG (15s) on Resonance Liberation.

Best Echo Sets
Looks like Wuwa is going to do 3-piece echo sets from now on, which means you can pre-farm the electro 3costs.

• Crown of Valor (3-Pc) + Void Thunder (2-Pc): +6% ATK, +4% Crit DMG (4s, 5 stacks) on Shield; +10% Electro DMG.
  Main Echo: False Sovereign (+12% Electro/Heavy Attack DMG, 405% Electro DMG on Intro, 8s CD).
  
• Void Thunder (5-Pc): +10% Electro DMG, +15% Electro DMG (15s, 2 stacks) on Heavy Attack/Resonance Skill.
  Main Echo: Nightmare: Tempest Mephis (+12% Electro/Resonance Skill DMG, 405% Electro DMG, 25s CD).

Echo Stats

• Main: CRIT Rate/DMG (4-Cost), Electro DMG/ATK% (3-Cost), ATK% (1-Cost).
  
• Sub: CRIT Rate/DMG (★★★★★), ATK% (★★★★), Heavy Attack DMG (★★★).

Best Team Comps

• Premium: Augusta (DPS), Iuno (Sub DPS), Shorekeeper (Support).
  
• F2P: Augusta (DPS), Mortefi (Sub DPS), Verina (Support).

Edit, Augusta universal damage amp makes her suitable for a lot of teams. You can just use two main DPS even.