not sure if you ran your game yet or not, but #1 - hope it goes well - #2 - trenloe's immediate response is pretty thorough - if you're solely running the module then TECHNICALLY you don't *have* to have anything else, but the core rulebook is obviously a pretty useful buy - the apg and character guide, again as mentioned, good resources for more character options, and gods and magic is a fun resource for informing your players on the content of the world, but not really mechanically important. The GM's guide, which isn't available yet on FG anyway, is actually more a huge collection of optional rules and not real mechanics adding to the current version of pf2 unless you want to start significantly altering certain gameplay aspects
I ran the first chapter of Age of Ashes as my first run on both FG AND PF2, and I think it's a solid module - you have at least 2 major npc's that can be really fun interactions for the players, a reasonably well-written cast of npc's for general RP, some okay combat encounters
Assuming you haven't run yet, here's some things to watch out for both FG related and module related that I ran into...
1 - a lot of the castle encounters adjoin several rooms - depending on how "flexible" you are with combat, you can have one monster retreat from one room and into others with other monsters, turning your combat into a mess of a running battle. You and your players may find this enjoyable, or you may find it mechanically super complicated and time-consuming - it's up to you, but it is an option, and possibly a problem, if your players are big on trying to find alternate ways into an encounter and end up triggering multiple encounters at once
2 - from a FG standpoint, the two biggest things I've noticed are getting your players to ctrl-click on a target BEFORE they start rolling dice. Players love rolling dice, and the automation in FG is awesome, but it only works if they've selected their target first. Also, try and emphasize early on to have players click the next turn button on the combat tracker - I'm running a 6 month open-world campaign and I'm STILL reminding my players to hit the next turn button...*sigh*
3 - when I ran AoA, FG didn't have the lighting effects up and running (or much of anything, really :) ) so no advice there, but assuming your ENTIRE PARTY doesn't build for low-light and/or darkvision like both of my parties have, those lighting effects could be fun - and important
4 - magic weapon (the spell) is a HUGE game changer for low level characters that don't have access to striking runes
5 - most encounters as written probably won't seem all that challenging to a group of players (although if your players are super new to PF2 AND/OR are just tactically inept, it may prove harder than normal) - most of my party steamrolled many of the encounters, and my general impression is that encounters are usually designed to "whittle down" the players rather than create a single challenging encounter in most cases
6 - using the medicine skill is a HUGE (I'd even go so far as to say VITAL) component of the game now. Healing up between encounters doesn't require spells or potions if you've got one or two decent healers and you give the party a 10 or 20 minute break between rooms - your perspective decides whether this is a good thing or a bad thing, but it does mean that spellcasters don't have to be heal-bots out of combat if they don't want to
those were my big takeaways from running AoA on FG - hope you enjoy it as much as I did...or more :)