Harmony considers collections of notes rather than individual notes. In practice, harmony is described by chords. It is very unlikely you would catch a musician asking "What is the harmony in bar 4?". Rather the question will be phrased "Which chord are you playing in bar 4?".
The chords that make up the vast majority of popular music, jazz included, can be divided into three main groups: major, minor, and dominant. We will keep a fourth group, a variation on the dominant called 'altered', on the side. Each of the three main groups is defined by the relationship between four notes: the 1st (also referred to as the root), the 3rd or the minor 3rd, the 5th, and the 7th or the minor 7th. The root and the 5th are the same for all three types of chords, but the 3rd and the 7th are not. It is the 3rd and the 7th that determine the quality of a chord. In a dominant chord, the 7th is flattened, and in a minor chord both the 3rd and the 7th are flattened. As you can see the patterns on the clock produce a line of symmetry in major and minor but not in dominant. Fortunately, you don't have to worry too much about that since it is possible to come up with 'safe' symmetric scales for all three types of chords. For now it is sufficient to make a mental note of the relationship between the chord notes in the three groups. Using the key of Ab as an example those relationships can be summarised as shown below.
In pop music you often hear chords played without the 7th. The open-string chords C, D, E, A, and G, for example, are made up of the root, 3rd, and 5th, and do not include the 7th. So are they major chords or dominant chords? By default you can assume they are major chords. Nevertheless, if the b7th occurs in the melody then the chord is a dominant 7. A good example is the first chord of the verse in "Can't Buy Me Love" by the Beatles. The tune is effectively a blues and the first chord is a dominant F7 but in transcriptions it is often written as an F-chord. If you play a major 7th on that chord it will clash badly with the melody.
The notation used when indicating chords on lead sheets and in fake books is far from consistent. If we include extensions and alterations as well the system becomes even more chaotic. For example, G6 is a major chord but G7 is a dominant. Furthermore, if you ask five pianists to play a complex chord such as G13b5b9 you are likely to hear five different voicings. Some of the most common variations, in the key of C, are listed below.
The bottom line is that you have to know which of the four groups a chord belongs to. The extensions and alterations are less important.
You transpose chords by rotating their notes on the clock. If you want to transpose Ab major 7 up four semitones you add four hours to 12-4-7-11 and get 4-8-11-3, which is C major 7. If you want to transpose C major 7 up another four semitones you add four hours to 4-8-11-3 and get 8-12-3-7, which is E major 7. There is nothing special or unique about any particular key. In terms of harmony, indeed in terms of music theory as a whole, it does not make any difference which key you start in. Only the relationship between groups of notes and the relationship between movements of notes matter. The only reason some keys seem to be preferred is because they are more convenient to play on certain instruments. C major is popular with piano players but Gb major is really no different.