Despite the fact that I've been writing music and performing melodies on my various keyboards for something like eight years now, I've only relatively recently started learning to play the piano in a formal fashion. You know: proper hand position, sight-reading, stuff like that.
Although I've had some musical training and could, in fact, read music, I wanted to learn piano as a beginner, as if I couldn't read music and had no training whatsoever. I did this so that I knew I wouldn't miss anything important. This might be beneficial for you too, if you're thinking of playing.
If you want to learn piano, here's what I think you should do:
First get yourself a keyboard. No one who isn't smoking hundred-dollar bills can afford a real piano, so don't bother. I tend to make fun of Casio keyboards a lot for their uninspired, out-of-date sound, but in fact I have a Casio keyboard in my bedroom and practice on it every day. The piano sounds fine on it.
Anyway, get yourself a keyboard that has a full 88 keys. Any less and you'll be hindering your own learning, and you don't want that. DO NOT get anything that has fewer than 88 keys. If you're really serious about practicing all the time, you will, within a few months, regret getting a smaller keyboard for cheaper. You'll start trying to learn an awesome new song that you've always wanted to play and then find that you can't because you don't have enough octaves to work with. This is maddening.
The best place to find a nice keyboard is a local music shop. They'll have a nice selection and can point you to a decent one in your price range. You can get one with 88 keys for around 200 to 400 dollars. It probably wouldn't hurt to check a place like Best Buy too, though their selection is limited.
Now you need to actually learn to play. You could pay for private lessons and they certainly would be nice, but you don't need them. There's nothing wrong with getting lessons, but you can very easily teach yourself. I've looked through a number of beginner piano books and the best one, the one that taught me to play, is the Hal-Leonard book Teach Yourself to Play Piano by Mike Sheppard and James Sleigh. You can probably find this one at the same music shop where you bought your keyboard.
Teach Yourself to Play Piano assumes that you have no experience with music whatsoever, so its a good start for the beginner. It emphasizes using the left hand as much as the right, it teaches some basic music theory, and best of all, it doesn't resort to godawful kid's songs like Mary Had a Little Lamb or London Bridge to teach you music. You even get to play a genuinely enjoyable and satisfying étude at the end of the book. You will be able to play piano by the time you finish, and it's only 50 pages!
After that, I suggest you find some beginner-level books of sheet music for you to learn from. Find books with music that you've always wanted to play, like from bands or composers that you really enjoy. This will make the learning process very rewarding. Not only will you be able to appreciate the songs on a more intimate and technical level, it's also simply very satisfying to hear yourself play a song that you love. I really enjoy music from video games, so I bought sheet music from old games Chrono Trigger and Secret of Mana, both of which have great soundtracks. You can also find a near infinite amount of sheet music on the internet, if you know where to look.
As you keep playing, you'll learn more and more about music theory. The best advice I can give you is: practice every day, even if it's only a few minutes. Really, you should be practicing about thirty minutes every day at least. For maximum results, practice an hour-and-a-half or longer every day and your skills will skyrocket in weeks.
Now that you've gotten to this point, take a look at my other article, Piano Playing Tips for Beginners.
By the way, no one paid me to endorse the book Teach Yourself to Play Piano. I don't get nearly enough traffic for anyone to pay me anything!