Well I would probably start with a best of. Probably '1' or the red and blue albums, (62-66 and 67-70 respectively). There's nothing wrong with starting with the first album as Jonathan suggests, but it will obviously take you longer, be more costly and not give you such an instant overview of their output.
Start with the red and blue compilations (62-66 & 67-70) and marvel at how far a band can travel in such a short space of time and take it from there. They really hit their stride from Rubber Soul onwards, personally, I wouldn't start with Please Please Me.
The only reason I hesitated to advise '62-66' was although it's a double, it's not particularly good value for money as the tracks are so short! It's a very good starting point though. Individual albums can be very personal, and some people will favour one over another. Most are enjoyable in their own way though. I always liked 'Revolver'.
Perhaps a compilation album would be better. I was bought the Red Album (62-66) on cassette when I was about 10 and it seems like a good place to start. I did think that it might be interesting for someone to discover The Beatles in the same way that the UK record-buying public did in the 1960s, i.e. in chronological order, but yes, it would cost a fair bit. I guess time may have taken away some of the excitement that accompanied the sound of the early Beatles and I know quite a few people who generally don't listen to their earlier work. Which is a shame, because the first two albums in particular (especially on the mono CDs, and hopefully the forthcoming mono vinyl), show just how good a rock 'n' roll band they were.
As a first generation Beatles fan, I would also recommend the Red and Blue albums as a starting point. You may find you might like a certain period of their music whether it be early Beatlemania (to 1965), Studio Years (to 1967) or later years (1968-70).
The true Beatle sound was Please Please Me album through to Help. The sound they produced afterwards is no different to others at the time or since (though great songs). The sound of the Beatles was the excitement and fun of the 62-65 period.