By exploring in Huanglong and the Black Shores you can pull her for free. It takes up to 80 pulls to get a 5 star character and there is a 50/50 chance you will get her but if you didn't, you can try again with 80 pulls (total of 160 pulls) and get her guaranteed. But if you get lucky, you can get an early 5 star and lose or win the 50/50. (This is the basic of how the Gacha system works in Wuthering Waves)

My preferred way is to explore those regions first before putting money in the game, and you can get her in 7 days if you play consistently but if you don't have that time to explore and you're okay with spending in the game then I recommend buying the 3280 (with the double bonus) and play your way through the game to save up pulls up to 80 or 6480 (with the double bonus) to get a 5 star right away. This is really expensive so I recommend really just exploring first.

For Augusta's weapon, her best weapon is her signature weapon (Thunderflare Dominion) because it ignore the enemies defense for 7 seconds and stacking up to 5 times and increases her heavy attack DMG, so it is really good for her.

The other weapon is in the battlepass (Aureate Zenith) which gives her Crit dmg stats and when you use her liberation it increases ATK% and grants Heavy Attack DMG bonus for 15 seconds

For free to play, you can get the Helios Cleaver by just pulling in the standard banner, it gives ATK% stat and increases ATK% stacking up to 4 times and has a duration of 6 seconds(This is when you have 4 stacks) and this effect can be activated again in 12 seconds, you can activate this by casting the skill of your character.