I don't remember a first experience with the band in particular, but I do know that I was mainly looped in by the one-shot Rupert cartoon (Rupert and the Frog Song) that Paul McCartney voiced.
But I do have other distinct memories, such as being allowed (once in a while) to play my mom's picture discs on my record player, and watching Help! in my mom's bed when I had a cold.
The Beatles really caught on for me when I was in high school though. I was in full-blown Monkees mode, and The Beatles just seemed to be the next natural progression. A friend of mine "loaned" me The Beatles #1 CD, and I played it over and over again in my $10 portable CD player (including one time during my art class, which resulted in the loss of all my participation points for the day, even though I did ask her first). I watched Help! and A Hard Day's Night to my heart's content, plastered pictures and 90s trading cards of The Beatles (mainly George Harrison) all over my room, giving them near-equal shares of space with The Monkees and Monty Python stuff. I spent nearly $30 of allowance money on an ill-fitting shirt off of Ebay. I begged my mom to let me keep her Beatles albums, and failing that, rented them all from the library and added them to the rotation of discs for my portable CD player.
There's a more complicated part of this story where I met a bunch of people through a vaguely related website, but that's another matter.
This isn't really an incredibly coherent history, primarily because The Beatles are woven into my personal history. Whatever people say about their musicality, or how they're over-rated, The Beatles have made me happy, and made millions of other people happy. I think that alone makes them worthwhile.
The Beatles left to right: Ringo Starr, John Lennon, Paul McCartney, and George Harrison.
And Your Bird Can Sing
Here Comes The Sun
I'll Follow The Sun
Love You To