The "Core Set" for Shadowrun really consists of Core, Runners Companion, Street Magic, Augmentation and Arsenal. Of the set Runners Companion is what I'd consider a must buy due to the sheer volume of metatypes, qualities and supplemental rules they pack into it for character gen and living in the sixth world. Street Magic is solid - it's much like the Grimmore from 2nd or 3rd edition : includes lots of spells, powers and magician specific qualities. It also deals a bit more heavily with the Astral world, Initiation and other concepts that were touched on in the Awakened World chapter of the core book. It also includes about a dozen or so traditions to help flavor your magicians. Useful but you can still run a frighteningly potent mage or shaman without it.
Augmentation has the requisite cyber goodies to make your sammy drool, and it fixes some bugs in the cyberware rules that existed in the original 4th edition rulebook from Fanpro: not sure if they're still extent in the anniversary edition. Also has rules for cyber-zombies, full body conversions and more. Worth it. The setting pieces about cybered life in the sixth world aren't to be missed either, covers lots of things hinted at in the core book's setting chapters.
I have all the above and I've gotten quite a bit of use out of them.
I don't own Arsenal but my understanding is that it's kind of a must if you want variety with your Riggers and Drone Operators. That said, it's also heavy on weapons and equipment. I passed on it way back when as I thought SR4 was already a bit equipment heavy for my taste, but running a game IRL now I'm thinking it might come in handy as I see how dangerous drones can be in the right hands.
As for the splatbooks it's a judgement call. I liked Runners Haven but I haven't picked up any of them since and I manage just fine, and I can't claim to have really used it either. I just . . .liked it. I would actually suggest the GM screen though - it includes a booklet including a random run generator which includes such perks as determining if the Johnson is going to screw over the party and why. Used it for an ad-hoc session and it worked surprisingly well. On the Run is only a decent adventure but I've canablized it for parts frequently enough that I can recommend it heartily. It has some useful pointers for first time SR GM's by one of the authors of the game as well. The other adventures seem fair to middling so far, but I haven't read all that many.