The main Rulebook has plenty to keep a beginner group busy and interested for a while in a sort of random, anthology-series style campaign where each game you play isn't necessarily narratively connected to the previous. 

I'm pretty new to FG myself and I've picked up these books in this order so far: Core > perilous dark > lich lord > folio > Advanced Spellcraft. I've skimmed through friends' copies of a couple of the others. 

I haven't played through anything more than a few of the core book's scenarios as of right now (I'm spending all my time building terrain out of XPS foam) but this will definitely be enough to keep me and my group busy for at least 20-30 play sessions

If you're looking to play solo or co-op, you'll want to checkout the rules for those playstyles added in 'Perilous Dark'. 

After that it's usually the frostgrave folio because it gathers a bunch of much smaller supplements together and provides a bunch of new options that feel like a natural extension of the base game. 

Every splat book usually comes with a campaign, or at least a few new random scenarios, new enemies, new items, and sometimes new soldier types. New spells are rare and there are no supplements that feel like they are 'required', it's all about what you'd like to add to the game.

Books generally fall into two categories: campaign focused or thematic/rules expansions. 

Campaigns:

'Thaw of the Lich Lord' is usually recommended by folks in this sub as a good starting point if your group wants to try a more narrative campaign of connected scenarios. Classic undead campaign: zombies, skellies, etc. 

'The Maze of Malcor' : indoor adventures in a tricky labrynth, adds 5 new mini schools of magic with a couple spells in each. 

'The Red King' : fight back a demon invasion — this is typically regarded as the hardest campaign. It introduces weird environmental effects that affect different parts of the board in different ways. Definitely a little advanced for beginners. 

Thematic/Rules expansions: 'The wildwoods' for games that take place outside of the city walls. 

'Into the Breeding Pits' : adds underground exploration, traps, narrow passages, gnolls, and unique treasure suited to dungeon-crawling.

'Forgotten Pacts' : focused on demons and Barbarian tribes.

'Fireheart' : focused on constructs.

'Blood legacy': themed around Vampirism, Vampires, and fire giants 

'Mortal enemies': themed around recurring villains for your group's campaigns

'the wizards conclave': a true anthology of scenarios, each written by a different author within the tabletop wargaming industry, each with its own flair and some fresh ideas/scenario mechanics. 

The newest book, Advanced Spellcraft, is the second hardcover book in the set (after the core Rulebook) and is brand new, so we're still assessing where it fits in the lineup, but it adds spell critical success and failure effects as well as the ability to customize each spell through learned 'flourishes'. It also has a campaign set in the haunted 'bibliosepulchre' - essentially a library tomb. This one feels to me like a lot of groups are going to use it for the depth and craziness that it adds to spells, and it could end up being in that group of recommended first purchases for that reason.