Decide whether you want to learn the songs using sheet music or tablature. Reading music is more complex than reading tablature, but it provides much more specificity regarding rhythm and dynamics. Tablature is the faster way to learn songs, but it is limited when describing notes that are not part of chords.
Pick a Beatles song and obtain the chords -- good songs to begin with include "Hey Jude" and "Yellow Submarine." Obtain the chords to the song that you pick -- the chords to any Beatles song can be found online easily. Once you find your song, look through the chords and make sure that you know how to play every chord in the song.
Practice playing through the song (play just the chords) until you are comfortable with the progression. Once you can play through the chords fluidly, play along with a recording of the song. This will help you with the timing of the chord changes.
Learn your first song well (try to memorize the chord changes so that you can play along with the recording from memory) and then move on to another song. Build up a repertoire of Beatles songs -- playing in public will further solidify your comfort with these songs.