I just DMed for my first time today. It was fun!

GOOD LUCK to yourself!!
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3 books:
The Player’s Handbook
Dungeon Master’s Guide
Monster Manual

A set of dice (not the same as regular dice).
4-sided
6-sided
8-sided
10-sided
12-sided
And 20-sided

A lot of paper and pencils. All this can run you between $60-$120. If you’re on a budget, and you’re very creative, you can get by with just the player’s handbook and the dice, although your game will be very limited. Alternately, you could get the dice and buy "Keep on The Shadowfell", a D&D adventure book that contains a basic rules summary, and does not require you buy the books to play. No matter what, you will need D&D dice.
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You would need at least two set of dice, the three core books, pen, paper, and graph paper. But if you find a group that will teach you and share their books all you need is pencil and line paper (or not)

They also have a startup "kit" that gives you a small taste of it with out spending a lot of $$ if you group doesn’t have any books. I’m think it is about $29.99 I suggest this product, but I don’t think they have one for 4th Edition yet and I’m not talking about the Player Kit.

To find a group try school bulletins,library, search for groups on website, meetup.com, or do a post at other D&D websites.
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Figure on spending about $100 or so to start from scratch. If you really get into the hobby with miniatures, extra books, and accessories you can spend as much as your budget allows.

The latest version is 4.0, just recently released, I think most people are playing 3.5 right now, but a lot will likely transition to 4.0. You could look for used 3.5 version if you want to save some money.

I never recommend trying to learn to play from scratch. Those three books have a lot of rules to learn and trying to read an play is often confusing and might spoil the game for you.

I recommend if at all possible find a group that knows how to play and join them, even if just for a game or two. There are a lot of players out there that are willing to teach the game to new players. Try looking for RPGA sponsored games (http://www.rpga.com) at your local game store. The other advantage is that they will have all the books, dice, miniatures you will need to get started.

As for classes, the core books have: barbarians, bards, clerics, druids, fighters, rogues, sorcerers, and wizards. For 3.5 there are a lot of extra books already out there that have more classes but those are the base classes. Starting out you should just worry about the base classes and base books.
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Thirty years of playing D&D.

Regardless of the edition, the only thing you really need as a player is the player’s handbook (or PHB). I believe the D&D for Dummies book really only covered 3rd edition classes – 3.5 added quite a few more in the core books and more still in the various expansion books. The first 4E core book pared it back down to about 8 or 9, but there will be additional PHBs coming out (at least one a year) which will expand things out again.

Don’t have the books handy, but IIRC the original 3.5 classes were barbarian, bard, cleric, druid, fighter, monk, paladin, ranger, rogue, sorcerer and wizard. Most of the various expansion books (Complete Divine, Complete Adventurer, etc.) each added a couple more (like warlock, warmage, scout, beguiler, marshal, etc.)

4E’s original set of classes (likewise, IIRC) are cleric, fighter, paladin, wizard, warlock, warlord and rogue. The PHB2 to be published next year will probably include at least some of the base classes from 3.5
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http://www.wizards.com/default.asp?x=dnd/welcome
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http://www.wizards.com/default.asp?x=dnd/welcome

Heres a site that has them: http://tinyurl.com/6evcdm
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My friend has the core rulebooks for 4th edition.
I play 3.5 edition.