As with any instrument the 3 primary elements are melody, harmony and rhythm. Melody is the single notes that most recognize as "the tune". If I played the rhythm and/or chords to "Mary Had a Little Lamb" most would not recognize the song. The melody could also be any "lead" or solo (improvised or not). Harmony is another word for chords or chord 'voicings' and rhythm is not the beat, but the variations of note values (whole, half, quarter, eights, sixteenth, triplets etc.) within the context of the beat.
How do we get better at melody? Well one element of melody playing is being able to play a familiar melody 'by ear'. Another aspect of melody playing is being able to read music. Music reading has 2 basic levels. Level 1 is being able to 'figure out' how something goes by reading the notes on a music staff and finding them on your instrument. The 2nd level of music reading is 'sight reading' where as you read the note you play it immediately. Of course there are different levels a sight reading ability. Rhythms, speed, key signatures, etc. But for the sake of argument let's say that a decent level is being able to read music in the staff (without extremes in ledger lines) eighth notes, at a reasonable speed and in the most common key and time signatures.
Another aspect of single note playing would be scales (from which harmony and melody is derived) for use as an improvising tool. Improvising is a fundemental aspect of playing most styles of music ( the most notable exception would be Classical music).
Entry level scales would include the 5 penta-tonic and blues shapes/positions in all keys. Further up the ladder would be the 7 modes of the major, hamonic and melodic (jazz) minor scales continuing to the altered scales, whole tone and both the whole half and half whole diminished scales. Ironically the end result is to not be thinking about scales at all. The old 'learn it to forget it' axiom comes into play.
Harmony, or chords, is important for guitar players because we are only 1 of 2 popular instrument groups that can play chords. The guitar belongs to the lute family. Fretted strung instruments like the guitar, mandolin, banjo, etc. are in this group. Guitar is easily the most popular member of the Lute family today. The other musical group that can play chords is the keyboard family including the piano and organ.
Level one of chord playing could be described as knowing the most common chords in the open position. The open position is the lowest area of the guitar neck (near the nut). Most open chords consist of one or more open strings and tend to use 3 or less fingers. Here is a list: A, Am, A7, Bm*, B7*, C, C7*, D, Dm, D7, E, Em, E7, F, Fm, G, G7. (*denotes chords that use 4 fingers).
Level two of chord playing might be knowing some of the less common open positions chords like Asus9 , DMaj7, F/G, etc. This level could include the ability to play any of our triads (major, minor, augmented and diminished) from any root note.
Level three would be complete knowledge of barre chords. A barre chord is one in which one finger (usually the 1st) holds down 4 or more strings (holding 2-3 is called a ˝ barre). If you have dabbled in guitar you are probably aware of barre chords. They seem to be some sort of milestone or dividing point in guitar playing. Some players stop short of barre chords because they found them 'too hard'. But with these chords you can play any version of major, minor or dominant 7 chords and should not be so easily tossed aside.
A common complaint about barre chords is 'My hands aren't strong enough". I can tell you I have met big strong men who couldn't play barre chords and small young people who could. So strength and hand size are not a guaranteed ticket to barre chords nor is the lack of hand size and strength an impediment.
One thing that is an impediment is guitars (particularly steel strung acoustics) with action (the distance of the strings from the guitar neck) set too high.
When the guitar is not the case the most likely cause is left hand technique! Simple things like thumb placement and even application of strength across all fingers can be addressed by a qualified guitar teacher.
Level four would be the ability to play four part chords with any root note Amaj7, A+maj7 A7, A+7, Am7, Am7b5, Adim7. We might include the ability to play the correct extensions to these chords like 9's, 11's and 13's
Level five would be being able to to play any of the aforementioned chord types in at least 3 different places on the guitar neck.
Rhythm oh rhythm. If there was one universal problem that most musicians have it is rhythm. It really is the deciding parameter between good, great and genius musicians. How does a non-drummer work on rhythm?? The same way drummers do by studying it! How is time subdivided, what does good time and rhythm sound like and more importantly what does it feel like when you play it?