Strongly Advised Books
- Rule Compendium - This is the go-to reference for the rules in the game, and is the most up-to-date resource.
- DMG 1 - A very useful resource for GMs including special rules and some GM advice.
- Monster Manual 1 - A good resource of basic monsters, and also lets you get a feel for creating your own (pair with DMG 1).
Thereafter it breaks down to what you want to see in the game. Keep in mind there's two flavors of DnD 4e, both that play more or less well with the other rule wise (IE players can draw from either half):
- Original - What DnD 4e launched with. Player Handbook 1, 2, and 3 offer some options, but keep in mind that a handful of rules and wording from PHB 1 have been redone. Rule Compendium always bests Player Handbook 1 when they disagree.
- Essentials - A refresh of DnD 4e with the aim of streamlining, and simplifying character creation, and to make classes less strategic and more fun. Ideal for the more casual players. Check out Heroes of Shadow, Heroes of the Fallen Lands, and Heroes of the Forgotten Kingdoms and see what classes and races interest the players.
And for clarification: there's a LOT of class overlap between Original and Essentials, but they're unique all the same (Rogue vs Thief, Assassin vs Executioner, Cleric vs Warpriest), so you can get the same story roles for the most part with either flavor, but the two (or three) variations split between the two halves of 4e will play differently. And again, the two halves play well with each other, so some players can go for original, others Essentials.
I believe rule style wise Red Box is part of the Essentials line, so you can judge if you want more detailed/strategic character creation (original line) or continue in the more light weight manner (Essentials).
I would NOT advice buying a ton of books starting out. Get a few, see how you like them and what you feel like you’re missing, and expand from there.
Edit: Also, if the players are looking for more of a challenge in their fights, Monster Manual 3 has monsters tuned to be more powerful and interesting for their levels.