Minimum, I would get the player's handbook and gamemastery guide in paper. For the rest you can use Archives of Nethys just fine.

As far as content, I would include all (some more) Archtypes from Archives of Nethys as the PH only included class archtypes and non-class archtypes I would consider essential for a better experience. I like Free Archtype and include it in almost all my games (sometimes restricted, somtimes not) as well. Using only PH classes and races is fine, but including everything in AoN won't break anything either.

Personally, I think the APs are all ok but not great. There are too many multi combat 'dungeons' and filler combat for my taste. That said, Strength of Thousands bones are pretty good and could be made great with some GM work.

The Mwangi Expanse is an amazing sourcebook but of course not essential. If you are at all interested in the region, I would pick it up.

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