It's a very long and winding road from Please Please Me to Abbey Road and it's hard to believe it's the same band only 6 years later.
In general terms you could say there's the Beatles before and after hallucinogens - basically the years of Beatlemania 63-66 up to and including Rubber Soul.
This is still the favourite period (and the lasting image) for many people when their lyrics were straightforward and generally upbeat. The 1964 A Hard Days Night album might be the peaking of this Beatlemania phase.
From Revolver onwards their music become increasingly more experimental and introspective and this probably peaks with the trippy Sergeant Pepper (Lucy in the Sky, Benefit of Mr Kite, Within Without You, Day in the Life) and the White album (Revolution 9, Happiness is a Warm Gun, Long, Long, Long etc).
My personal favourite is Abbey Road because it's the one I'm most likely to play without skipping from start to finish.
Side 2 (on vinyl) for me is a good a side of music as there has ever been, and side 1 is not too bad either.
As for the concert/ touring years (62-66), I think Ron Howard's Eight Days a Week DVD is great place to start.