It should go without saying that as DM you must be familiar with all of the rules. That doesn't mean you should be expected to quote any given rule on demand. Read the books… many times. Make sure you understand everything presented. Make notes and do your best to commit the most important parts to memory. But your players will understand if you forget something. They will usually be quick to remind you. Plus you have the books there to look rules up as need be.
You may find it difficult to read the books from front to back though. Every other paragraph may mention something that you aren't familiar with and want to look up. Go ahead and do that. For this type of material, I think that's probably the best way to learn it. Skip around. Read a section at random and bypass sections that don't sound so interesting right now. Once you have a basic understanding of various aspects of the game, go back and read each section again in order. Hopefully that will help make it all make sense.
When you get to the point where most everything is clear and you have a firm grasp of the combat system and the various options available to players during combat, go back and reread the powers and feats. When players are in dire straits and looking for a loophole, they will come up with some interesting interpretations of the rules. You don't want to be caught off guard by a feat or power you don't understand.
Another section to pay particular attention to is the Magic Items descriptions in the PHB. A powerful magic item in the wrong hands can unbalance the whole game. Once you give it out, there is no way to take it back without causing some hurt feelings.
The Monster Manual is basically a book of stats on each of the monsters. Nobody is going to be able to memorize that! What I think might help though is knowing which monsters have special powers that could be particularly dangerous in certain situations.
Imagine if all the members of the party have favorite magic items or spells that each do fire damage and they run into a group of monsters that are extremely fire resistant. What should have been a balanced fight could easily become a party wipe. You could argue that the party should be more careful to balance their damage types. Then again, being right is small consolation when the players get mad and don't want to play anymore.
Lastly, look carefully at new rules, or alternate rules, before adding them to your game. There is a ton of source material out there from various books, magazines, websites, etc. Just because someone says it's a good idea doesn't mean that it is. evaluate for yourself how it will impact your game.