How to Play Jazz Piano

Jazz is an art form that has grown from its blues origins to draw influences from just about every genre of music there is. For the beginner, though, it's perhaps best to focus on early swing and learning to improvise. Here's a pretty easy way to get going!

Steps

This is by far the most important step in becoming a musician. Find as many recordings as you can get your hands on. Don't discriminate--listen to the old greats, like Art Tatum and Count Basie and Thelonious Monk, as well as up-and-coming pianists of today. Listen, take what they do, and apply it to your own playing. Doing this consistently will make you an excellent jazz pianist.
Assuming you already know some very basic theory, first learn all 12 major scales (there are twelve different sounding scales, but in theory B/Cb, F#/Gb and C#/Db are separate scales). Learning all the scales will be extremely helpful.^{[1]XResearch source}
The first real step in your journey will be to break away from "the dots" and train your ear. So...
Cole Porter, Gershwin, etc. Make sure that chord symbols or guitar tabs are written above the melody line, like "Dbm7."
Learn a major 7th (1 3 5 7), minor 7th (1 b3 5 b7), dominant 7th (1 3 5 b7), half diminished (1 b3 b5 b7), and diminished chord (1 b3 b5 bb7) of every key. So, for example, to play C7 (C dominant 7th) you'd play C, E, G, and Bb. For C diminished seventh, you'd play C, Eb Gb, and A (Bbb). You need to know them well enough to be able to see a chord symbol like the one in the step above and be able to play it without thinking. If you know your major scales, you could have this step mastered in a week.^{[2]XResearch source}
Find a song you like and play the melody line in the right hand with the appropriate chords in the left, as you're reading them from the chord symbols. You are now playing a song without reading music (in the traditional way, aka Fakebook style). Congratulations!
Even though it probably sounds horrible, practice for long enough and you'll sound more and more like what's written there without you even knowing it. You can always go back to the sheet music to see how they're voicing the chords in clever ways that you're not.
learn to play CM7 like (C, E, G, B), (E, G, B, C), (G, B, C, E) and (B, C, E, G). Learn those four positions for every chord, but only after you're comfortable knowing what every chord is, and have Step Four under your belt. Don't scramble your brain.^{[3]XResearch source}
Add some more, and take some of the originals out.
By now, you're probably IMPROVISING! Learn those two scales for every key.
Try to splice one from one song into another.
Also learn tritone substitutions and the circle of fifths. Play the same songs in different keys.^{[4]XResearch source}
Learn modes and different scales. Listen to different sorts of music from all sorts of time periods, and anything that you can steal harmonic and melodic ideas from. When you've gotten this far, you can easily teach yourself.^{[5]XResearch source}
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Should I aim for a certain amount of time to accomplish each step?
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No. Take as long as you'd like for each one. A few of them don't have a time frame. For example, listen to the greats your whole jazz career and beyond. Listen to them in the car, while you are studying, etc. Each and every song, no matter how many times you listen to it, will offer knowledge and understanding, whether you are just starting out or are an established musician.
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Do I need to be a sight reader to become a jazz pianist?
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No. You do not need to be able to sightread music. But you should have a good sense of rhythm and swing, and unless you can play by ear, you need to know how to read music.
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What's the relationship between blues scale and pentatonic scale?
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Here's an example. The A minor pentatonic scale consists of the notes A, C, D, E, G. The A blues scale adds the chromatic pitch in between D and E. The whole scale is A, C, D, D♯/E♭, E, G.
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Tips

Do not forget: you learn to play the piano by playing the piano, not by reading a book or a How.com.vn. You learn to do things by actually doing them. Experience is everything. You want your hands to know how to play, not necessarily your brain. Take a small step at a time to master a piece, so that you can really take in the notes and technique you're playing.
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Don't be afraid to make wrong notes, sometimes it might even sound good!
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Love jazz, and learn to love the craft of songwriting. Listen to jazz music.
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