Hi! Welcome to the hobby and welcome to the boards.
Everyone needs the PHB (Player's Handbook), including the DM. It is the most important manual by far; very expert people can do a lot without anything else but the PHB is a must.
The DMG (Dungeon Master Guide), the book which you bought, is an additional guide for the DM. It contains lots of stuff that the other players don't need to know at the beginning, such as how to design a good adventure, or what damage the environment can do to you. Some stuff in the DMG becomes useful for players as the game progresses (prestige classes and magical items, mostly).
The MM (Monster Manual), the last core book, is simply a big bunch of monsters (plus some rules to make your own). Since D&D has a
lot of weird creatures, they have been described in a separate book. The PHB only contains the races which the players can choose for their characters.
There's a lot of good advice you'll probably get here, I'll limit myself to: learn from your mistakes and you'll get good quickly.
