The melodic minor scale is commonly used by jazz musicians. There is only one note (the third) different from the major scale. It has also been used a lot in traditional and classical music.
Please refer to the minor scale article in Wikipedia if needed.
On the guitar it can be played in using open strings :
Also, it is highly common for jazz guitarists to sing along to their playing. Try it! It helps to reinforce the link between your fingers (what you play) and your ears (what you hear inside).
The theoretical side of the melodic minor scale is easy to grasp. It is built of seven notes that are laid out using intervals. The half-step interval is one fret away while the whole step is two frets away.
So the formula for the construction is :
W H W W W W H
(W stands for whole-step and H for half-step)
The numeric formula is : 1 2 b3 4 5 6 7
(it means that in comparison to the major scale, it has a b3)
To me, the melodic minor should be learned immediately after the major scale. They have six notes in common!
Please see the "Theory" section of this website for more theoretical details. (Chord construction and chord progressions.)
It is a very good way to "compartmentalize" the neck. You will get familiar with the whole instrument dealing with it chunk by chunk.
Once comfortable, move on to the jazz guitar scales. There, you'll learn how to expand single string and position playing for improvisation.
Jazz Guitar Chords and Jazz Harmony