It is the greatest misfortune that most people in this country are introduced to Shakespeare in the worst possible way; usually as a boring high school assignment, and usually a tragedy (I personally had Romeo and Juliet, Othello, and Julius Caesar in HS).
Other than the general run of HS reading assignments not being all that interesting or useful, and turning even the best works into unpleasant experiences; often what hangs people up with Shakespeare is the language. It definitely takes some getting used to (though not nearly as much as Chaucer - another great author of "common" literature butchered by HS English).
Combine the language issue, with an unfamiliarity of the history, expressions and idiom, and dramatic or literary conventions of the time, and you end up missing a lot of what is truly excellent, and honestly generally not at all difficult to understand literature.
First note, these were written specifically AS PLAYS for the general public (the English "middle class" as much a misnomer as that is) which means they have an inbuilt natural narrative, flow, and pacing that should be a joy once you are into it.
Anyway, back to the question, how do you start out?
Well, I recommend you start with a comedy not a tragedy. The comedies are much lighter, and more quickly engrossing.
Much Ado About Nothing is an easy intro, and my favorite of the comedies. twelfth Night isn't too bad either.
My favorite play by the way (from anyone, ever, not just Shakespeare) is unquestionably Henry V, but it's not one that's really suitable for beginners (nor any of the histories really).
Measure for Measure is not his best work, and the language is complex, but relatively easily understood (it uses simple words in complex constructions, and as such is also very quotable). As you like it could be regarded similarly. The taming of the shrew is a better play, and also entirely palatable.
In terms of the tragedies, Hamlet, Lear, MacBeth, and Othello are such must reads that you can't bother about the difficulty (if indeed you find them so. Lear is a bit thick, as is MacBeth, but I find Hamlet and Othello quite compelling). One supposes one MUST read Romeo and Juliet, but I honestly think it far inferior to the others. Actually my favorite parts of the play are those dealing with side characters (Tybalt is an interesting little psychopath, and Mercutio gets the best lines of the play).
Finally, you absolutely have to read Richard III and Henry V. Yes both are drastically historically inaccurate, but both are absolutely brilliant.
So my list of Shakespeares "must reads" is I suppose thus:
Comedies:
Much Ado About NothingTragedies:
Twelfth Night
Measure for Measure
As You Like It
The Tempest
The Taming of the Shrew
The Merchant of Venice
A Midsummer nights dream (I dont like it much, but it's culturally significant)
OthelloHistories:
Hamlet
King Lear
MacBeth
Julius Caesar
Romeo and Juliet (again, I don't much care for it, but you HAVE to read it)
Richard IIIThat's my must read. I don't much care for the sonnets, but that's just me. Oh and I'd stay away from Titus Andronicus, it's not only NOT a good play, but it's hard to read, poorly written, and frankly disgusting.
Henry V