My choice of 3 products would be the Core Set and then either/or the Adventurer's Kit and Signs of Faith.
You obviously need the core set for all the bits - and it's the most economical way to buy dice too.
The Adventurer's tool kit is one of the cheaper expansions, and so bang for buck will give you more careers, and more 'stuff' for the players to take to customise their characters: actions, talents, etc. There are quite a lot of good ones too. I think it also includes basic cards for a 4th player, if you want to use these (I would rather remove all of the basic action cards for the sake of table space and replace them with one of the downloadable summary sheets.
The Signs of Faith expansion again has some cool careers in it, plus the disease rules. Of all the things 'missing' from the core, I think these are probably the most important to the setting and difficult to house rule in a satisfactory way. The other reason I like this set is that the background on the gods and religions is important information for fleshing out the world for players. Whereas the colleges of magic background is really only relevant to the GM and wizard players, most characters should be familiar with all of the religious background. You did mention though that you didn't need any background…
I personally would not recommend the guide books as essentials if you're starting out and looking to avoid spending money needlessly.
The player's guide is okay, but obviously repeats a lot of information you have in the core set. It may be clearer, but I think you can download rules summaries / cheat sheets / alternative GM screens which will do this just as well. If you were to buy Signs of Faith you would additionally have covered half of the expansion material in the book that you don't get in the core set.
The monster guide will be more or less useful depending on what kind of game you want to play, and how comfortable you are at winging things. In my view it fails at providing an easy-reference 'everything you need to run a particular monster on one page'. Information is spread out between different chapters, cards, rule books, etc. You'll have all of the monster background from previous editions, so presumably won't need all that again. Many of the actions are really very similar (is a 'claw strike' really very different from a 'sword strike' from a 'pointy tooth strike' from a 'horn strike'?). You have some core monster rules and actions in the core set, and in my opinion they're plenty to be getting on with. If your game will concentrate a lot on human opposition - thugs, cultists, soldiers, wizards, the city watch, etc., then you won't need lots of new monsters. The core rules contain many of the core monsters, and while you may be missing 'lesser spotted banshees from Talabecland', are you ever likely to put one of them in your game anyway? And if so, can't you come up with some cool special rules yourself?
If you get anything else, I would probably go for the Omens of War - for the severe injury rules and more cool careers and fighting options, but might not worry if your players aren't very interested in tricking out their characters for combat. If you have wizard PCs, then the Winds of Magic is probably a must, but if not, it's an awful lot of content that really won't be used in games, as cool as the careers are. I think the Lure of Pleasure 'extras' are less important than some of the others (social combat and secret societies are less in need of rules mechanics than physical combat, disease and/or magic), and I don't think the adventure is that brilliant, but there is some good setting material, and again, the careers are pretty cool too.