The choice of chords in a song is largely dictated by the key signature used and applicable movement on the Circle of 5ths. The Beatles learnt to compose and play music without reference to much any music theory, but their progressions DO fall within the limits that are part of the Circle of 5ths. Helped by their producer who was trained classically as an oboe player, some of their knowledge can be attributed to him, but they did copy music that had well established patterns -- all within the IV- I - V area. They tended to break rules just as much as any other artist. For those interested in their music, it has been analyzed many times. The analysis by KG Johanssson is a good reference to their work. They tended to use the Key of E and the Key of G.

Songs in the Key of E often use the following chords: E (I), A (IV), and B7 (V)



The next chords that one should learn are known as Secondary Dominants and this is where I will leave you. Wikipedia also has information related to these types of chords. Last of all, those interested in teaching compositions should check out Mozart's K. 545 Piano Sonata No. 16, written in 1788. It was not published until 1805. It has three movements (Allegro, Andante and Rondo). An analysis of Sonata can be found at Brandy N. Gerber's Musical Insight webpage. Audio of the 1st movement can be found on You Tube.