Jazz Piano I
One of the first things a jazz pianist needs is a foundation of good pianistic technique, which is most easily gained by studying classical piano. Learning pieces by composers known for using counterpoint, especially J.S. Bach, can be especially helpful in developing an ear for contrapuntal melodies and hand independence (the ability of both hands to do different things at the same time), which are both critical to the jazz pianist. Several of the technical resources used in jazz piano are classical devices which you can find in the baroque era.
In order to successfully improvise, a correct articulation is necessary. Although this can only be achieved through practice—classical training may help—, the best advice is to "feel" the music you are playing. Technical aspects involved in rhythm, for instance, are likely to be better understood if they are felt and thought rather than just read from sheet music. In jazz, as a rule of thumb, most passages are played with eighth notes, in a more or less "swinging" feel. But to learn rhythm patterns, it is usually better to do it by listening.
By collecting a library of good jazz recordings and listening regularly, you'll be able to determine which pianists you might like to emulate, familiarize yourself with the myriad sounds that constitute "jazz", and develop an ear for the common forms jazz tunes tend to follow (AABA, for instance). This will also give you an idea of which tunes you would like to learn first. There are numerous great pianists that you will discover in time, but the (almost) undisputed first six on your list should be Art Tatum, Bud Powell, Oscar Peterson, Bill Evans, Thelonious Monk and McCoy Tyner.
Learning Tunes 
Before learning a lot of complex scales or chord voicings, it is essential to learn some of the more common tunes jazz players play. Here are some examples of tunes you'll want to learn:
- Autumn Leaves
- All the Things You Are
- Beautiful Love
- Body and Soul
- But Beautiful
- Darn That Dream
- Fly Me to the Moon
- Girl from Ipanema
- I Hear a Rhapsody
- My Funny Valentine
- There Will Never Be Another You
Resist the urge to read tunes out of the book and memorize them as soon as possible. It is better to buy a recording of the tune and learn it from the recording. It is critical that you feel relaxed and free to use your ears to explore possibilities in a piece of music, and you'll never accomplish that while you're reading it from a page.
Start learning each tune simply, beginning with memorizing the melody and also the lyrics, if there are any. One good way to learn a melody is to play the bass note of each chord symbol while singing the melody. Later, replace singing the melody with playing it in your right hand, but sing along softly or silently in your head. To ensure you really know the melody, try transposing it into two or three other randomly chosen keys.
Next, fill in the space between the bass notes and the melody with the inner chordal material indicated by the chord symbols. Here it will be useful to talk about the chord symbols you are likely to encounter.
Blues Scale 
C Eb F F# G Bb C
Play these as eighth notes with a slight swing.
I would like to state that the displayed scale is a blues scale rather than a "jazz scale". There is no one scale which can be called the "jazz scale". Jazz music uses many many scales, which belong to the various groups of modes.
It's a minor C blues scale used when playing a blues in C (major). The same scale can be used as the equivalent major scale, that is in this case, Eb. Try to take the Eb chord with your left hand and play (or try improvising in) the above scale. The bass note is now of course the second one - Eb. You will see that this major-6 scale is a very good jazz scale and probably more useful than the C minor in most jazz tunes. The same scale in C would look like this: C D Eb E G A C
Adding an Ab to the C6 scale gives you another blue note between the 5-6 (G-A). These scales in theoretical terminology are known as the pentatonic scales. Many improvisations made by some pianists give this pentatonic feeling as the key to improvisation. Something else to consider is that parts of chromatic scales are often used when improvising as well.
Using Appoggiaturas and Other Melodic Phrases 
In music, an appoggiatura is a note that is played quickly and chromatically, for example in the C major scale a pianist could play C, D, D# E, where D# becomes the appoggiatura and is played right before E. In most blues piano improvisations there is a famous melodic phrase, used before the fifth chord on a chord progression is played, called the Huey "Piano" Smith turnaround, named after the pianist who invented it in New Orleans. Most of the music containing good improvisational phrases and piano soloing also comes from this city. This turnaround in a C Major scale would contain an appoggiatura, thus the following notes are played with right hand: C D# (E and G simultaneously) C (F and A as an interval); C, D# (E and G simultaneously once more) A (crossing index over thumb, which falls naturally into G and then G is played) More concisely, the turnaround repeats the first interval with the appoggiatura and notes F and A are played with fingers 4 and 5 respectively. Another interesting phrase used in piano for jazz or blues which uses the black keys to form a quick blues (pentatonic) scale can be heard in the song "Big Chief" by Professor Longhair.