I'm a beginner to DnD, and I've been reading everywhere that you should buy the Red Box Set, but this is about $100, in comparison to the DM Kit, Monster Vault, and Hero's of the Forgotten Kingdom (Which has a supplement to the PHB Correct?).
The 4th edition Red Box is out of print. The $100 prices you see is basically what it is worth as a collectible, not the original price tag that Wizards of the Coast put on it. While it is a nice introduction, you don't get very far with it, so you are probably better off if you buy the other three products you mention.
I, as a DM do not think that you need the red box. I think that it is just (another) money making technique that they use. I think you only need Phb and Dmg (players' handbook and the dungeon master's guide) in the respect of books. The DMG only needs reading once or twice but the PHB I recommended having copy in some form (physical or virtual). Happy playing!