Many would-be pianists have told me that they stalled out on the piano because they just couldn't get their two hands going at the same time. "First I learn one hand," the story goes, "and then I learn the other, but I can't stick them together!"
The thing is, that's not how pianists think. Flute, saxophone, trumpet - these are instruments that can only create monophonic (i.e., one note at a time) textures. The very nature of piano is polyphonic (many voices), and most of the music written for the instrument takes advantage of this.
But polyphonic does not equal "monophonic plus monophonic" - you can't just learn the parts separately and mash them together. You'll need a different strategy.
Playing the piano - an integrated process
Playing with two hands is but one process. It's like driving (hat tip to Michael Rachap for this insight). You can practice your blinker when you're stationary and you can practice working the clutch in the parking lot. But the only way you can practice shifting is to take the car out on the road. Even if you're driving an automatic, it's an integrated process - boiling it down to its individual elements isn't going to help you with anything.
The juggler has a job to do with each hand, but they can't really be practiced separately - you're always tossing the ball from one hand to another. The guitarist can't play one-handed - the right hand has to pick or strum based on what the left hand is doing. The secret to coordinating your hands to play piano fluently is to see it as a similarly integrated process.
Keep the hands together from the start
Sure, there are times when a section played by one hand is so challenging that you will have to spend a lot of time ironing it out before you proceed. But past the point of basic familiarity, continuing to practice them separately will not help you to put them together except to the extent that jumping higher and higher will allow you to fly.
The solution on a given piece is to take smaller sections, hands together. Working from the end can be helpful. To read the music effectively, you have to be able to see the musical patterns in the song. If you're new at this, working on easier repertoire will help.
The grand staff - two staves in one
It is helpful to visualize the two staves (bass and treble) as the one thing they are: the grand staff. Resist the temptation to separate them, even mentally. The brace on the left side of the grand staff is code for "This is piano music. These staves are inseparable."
Imagine the music as a series of "events"
In the example at left, there are three notes that happen on beat one of the first measure, followed by three notes in the right hand. Then, there are three notes at once on the first beat of the second measure.
It does take a moment to read all three notes if you don't have a ton of experience (that's why working on easier stuff is so helpful). So let's eliminate the notes for now and instead think of them as a series of events happening in time (rather than as a melodic line).
Here's how to practice that: place a hand on each of your thighs. When the left and right hand play together, slap your hands down on your thighs simultaneously. When the right hand "plays" by itself, slap just the right.
The example above would go like this (where X means "slap your leg"):
Here's another example, an excerpt from Haydn's Piano Sonata No. 40, with the corresponding "leg version" below it:
Obviously, that's an oversimplification. In fact, in the last measure, there are actually two voices to be played by the right hand. To learn this most effectively, you would practice the bottom voice of the right hand with the left hand, since they go together. Then you would layer the top voice of the right hand on top. In other words, separating the hands isn't going to help you, but separating the voices might.
In any case, even though this is simplified, it will be helpful to you if you've never thought about the right and left hand parts as an interlocking series of musical events. Once you understand that concept, you just need the specifics.
Making it automatic
If you're not reading from a score (in other words, if you're playing by ear), coordinating the hands comes down to muscle memory. In certain styles, such as blues and boogie-woogie, you'll need to make sure the left hand is automatic before you layer any right-hand patterns over it. Once the left hand is automatic, this will be relatively easy.
In pop and rock styles, visualizing the "events" as described above can be helpful, as many grooves are a combination of left and right hand movements.
Think like a pianist
My goal with this post has been to help you think like a pianist, visualizing the two hands as parts of an integrated whole. As you progress, though, you'll find that the hands combine with a seamlessness that is as easy as singing.