"How High the Moon" is one of those standards that has been covered by so many artists, it is tempting to think it is an easy song.  Indeed, the melody is very straightforward.  And the song form is a very conventional 32-bar ABAB.  And there aren't really any strange chord changes.

But don't be deceived.  While the changes seem very natural, this is a slippery song.  In particular, it is loaded with cases where a major7 chord is followed immediately by a minor7 chord on the same root note.  That major-minor exchange sequence comes up somewhat regularly in the American Songbook.  When you see that, you can be almost certain that the key center is moving down one whole step.

Why is that?  Let's take a look at the first 5 measures (in concert key):
    Gmaj7     Gmaj7    Gm7      C7      Fmaj7
The first two measures are simply the main key center of the song: G major.  In bar 3, we go from G major to G minor.  That Gm7 begins a ii-V-I sequence in the key of F.  So as soon as we hit that that minor7 chord, we are immediately in the key center one whole step down.  And that major-minor exchange continues throughout the song -- each time lowering the key center a whole step.  We see the exchange happen on bars 3, 7, 18, and 22.  Each of those ii-V-I patterns is 4 bars, so exactly half the song is part of the major-minor exchange pattern.  The key point is to hit the new key center on the ii chord (bars 3, 7, 18 and 22), not a couple of bars later.

There is another complication that makes this song tricky.  The A sections are identical.  The B sections are almost identical, with the big difference being the minor chord (Gm7) in bar 11 versus a major chord (G maj7) at the same point in the second B section at bar 27.  With that being the only real difference from the first AB to the second AB, you must clearly hit the minor the first time and major the second time, as that really defines the whole song.

One final point of trivia with this song.  It has essentially the same changes as Charley Parker's Ornithology, although the melodies are radically different.  This arrangement superimposes the Ornithology line on the last chorus, just for fun.

Do you like the song "How High the Moon?"  So did Charlie Parker.  He wrote Ornithology using exactly the same changes as "How High the Moon."  Parker's melody is completely different, and usually played at a blindingly fast bebop tempo.
The title of this song is tongue-in-cheek.  Charlie Parker's nickname was "Bird" and ornithology is the study of birds.
The biggest challenge of this song is probably the tempo.  This arrangement is set at 190, which is quick, but the song is often played at 220 or higher.  The song form is a fairly conventional 32-bar ABAB, although the B sections are not identical.
If you find it overwhelming to improvise these changes at this speed, don't despair,  Just practice the head until you can play that at 190.  Try practicing the improvisation on "How High the Moon".  That will be easier because it is a more relaxed tempo.  The commentary on "How High the Moon" includes tips useful for improvisation.

Why does it seem like every other jazz standard is written by Antonio Carlos Jobim?  Because he was a stinkin' genius.  He combined interesting rhythms, beautiful melodies, luscious harmonies and cerebral lyrics.  A perfect example of all these elements in Triste.  This song should be in every player's bag of tricks.
Like most Jobim songs, Triste takes an excursion through various key centers, but does it so elegantly that the player can easily be tricked into thinking this is simple harmonically.  It isn't.  You really have to learn the melody and the chords internally.  Nearly half the chords are part of a ii-V pattern.  Whenever you see a ii-V pattern, you can consider it to be a single chord, so the chord changes don't come as quickly at may appear at first glance.
The song form is a straightforward 32-bar ABAB, except that bars 31 & 32 are usually repeated, making a 4-bar vamp, or a total chorus of 34 bars. 

This Miles tune is a simple "almost" blues.  It is mostly the familiar 12-bar pattern, but notice that the song form is a double-chorus.  And most importantly the first chorus does not end up on the root chord. 

The song is in Bb concert, so you would expect bars 11 and 12 to be some flavor of Bb, but instead that first chorus moves down to Ab7.  The soloist (and band) must keep track of the even and odd choruses to make that distinction on the even choruses.

Here is another Horace Silver classic, Nica's Dream.This has a harmonic minor vibe (minor with a major 7th) vibe throughout.  In this case, it is built around the concert Bbmi(Maj7) chord, with that scale being:

 Bb - C - Db - Eb - F - Gb - A NATURAL - Bb

Practice that scale thoroughly before trying to improvise on this tune.  It is an awkward key (5 flats concert) for wind players.  This is another of those songs where the bridge is more conventional than the A sections.  Simple AABA 32-bar song form.
This song is often done in swing , or with the A sections in Bossa and the B section swung.  In this arrangement, we stay in the Bossa pattern throughout.

"Nightingale" is a popular jam session tune.  It is a lovely tune with interesting harmonies. This song is often played (sung) as a balled or medium swing.  But it can also be played straight or up-tempo.  This arrangement has a fusion vibe, and we've done some aggressive re-harmonization to give this old classic a modern life.
The song is rather straightforward.  It is AABA song form.  The A sections are each 10 bars.  The bridge is a very conventional 8 bars.
Enjoy.

Here is a great tune for a lazy summer afternoon.  There are many jazz standards where the music was written first, then years later after the melody became famous, somebody wrote lyrics.  The lyrics to this song are so sympathetic to the melody that is is hard to believe they were not written together.  Thad Jones wrote this beautiful melody.  Years later, Alec Wilder wrote the lyrics, evidently not as any organized collaboration, but they fit perfectly:
   Now, out of the night, Soft as the dawn, Into the light
   This Child, Innocent Child, Soft as a fawn, This Child is born
   One small heart, One pair of eyes, One work of art, Here in my arms,
   Here he lies, Trusting and warm, Blessed this morn, A Child is born.

This is a simple ABAB song form of 32 bars.  The only complication is that conventionally, the solo choruses use a 30-bar form, leaving off the the final two bars.  In this arrangement, all choruses are a full 32 bars.

Miles Davis played so many great tracks that it is hard to say what song is his most notable.  "Four" is one of his most popular jam session tunes.  It is a simple ABAB 32-bar song form except for an optional tag to the final chorus. 
The harmonies aren't terribly difficult once you get settled with the ii-V-I progressions.  This is probably the most common chord sequence in all of music.  Composers have used this pattern almost since they began writing music.  Classical musicians may prefer to call this the sub-dominant, dominant, root sequence or something like that.  But the concept of "root" doesn't apply so much in jazz because people like Miles used the ii-V-I pattern as a way to move gracefully between otherwise disconnected key centers.
We find ii-V in the A section in bars 3-5 to get us from Eb major to Db major. In the B section we have the ii-V (of E) with the F#m7 - B7, but in this case, Miles takes us not to E major, but to F minor.  It is unexpected, but elegant.  The important thing about these patterns is to make sure you land strongly on the chord after the dominant.  If it is a true ii-V-I, that is easy because all three chords are tightly related, using the same notes.  But if the landing chord is not the expected "root" chord, you must prepare for the surprise landing.
In reality, the song isn't as complicated as all that, and you might do well just to become very familiar with the melody and let your ears guide you.

An old classic, Bésame Mucho, arranged in a Brazilian bolero style.  This is a relatively easy song for improvisers because the song remains in the same key center throughout.  There is no crazy bridge that goes off into some distant key.
The song form is ABA with the A sections having 16 bars each.  The bridge is 8 bars, giving a total of
16 + 8 + 16 = 40 bars.

Benny Golson is a living legend, one of the few musicians alive who has not only seen the entire modern jazz era, but actually played a major role in the development of the art form.  Although he is relatively unknown outside the jazz aficionado world, his influence has been immense.  As a composer, he is probably best known for "I Remember Clifford" and "Killer Joe".  But he has composed many great songs, including "Along Came Betty".
When you see the name Benny Golson on a piece of music, you will know two things immediately:

1. It is a carefully crafted piece with a catchy, memorable melody.
2. Even though it makes perfect musical sense and sounds like it might be easy, it is probably actually very challenging for the improviser.
   

That is certainly the case with "Along Came Betty".  The song form is fairly conventional.  One might call it AABA, although the second A section is only reminiscent of the first.  The B section is distinctly a bridge.  The final A section starts like the first A section but has extended phrases.  That gives us a measure count of 8 + 8 + 8 + 10 = 34 bars.
The changes are not complex harmonies, but they are less predictable than in many standards.  There are several places where the music shifts up 1/2 step and then back down.  These are important moments you find in measures 1-4, 9-12, and 25-28 -- or about 1/3 of the chorus.  To get started improvising on this piece, it would be effective to find a comfortable line for measure 1 and then play that same line up 1/2 step in measure 2 and so on, then depart from that simple pattern as you get more familiar with the changes.
There really are no great shortcuts on a song like this.  Just keep practicing it until everything comes automatically.
It is worth mentioning that this song is usually done as a medium swing.  This arrangement is a straight-8th smooth jazz treatment.