This is often cited as being Shakespeare’s worst play. T. S. Eliot went so far as to call it the worst play in the world. I’m personally not persuaded that it is either. Perhaps unsurprisingly, it plays much better on stage or film than on the page. It’s also probably worth noting that it was Shakespeare’s most popular play during his lifetime.
In order to appreciate the play at all, you probably need to have (or cultivate, at least for the purposes of this exercise) some appreciation for the Renaissance revenge drama. There’s a long tradition, starting long before Shakespeare and continuing well into the seventeenth century, of plays in which the primary motion of the story is the fulfillment of a vast and bloody revenge on someone who usually deserves it. This is worth knowing something about, since even if you never cross paths with Titus Andronicus again, it will help you come to a much better understanding of other things as well — Hamlet, among other things.
It also makes a good foil to some of the other late tragedies, like King Lear and Othello, and, at the most unexpected points, breaks out into Shakespeare’s lyrical and intense tragic verse of the sort we’re accustomed to attribute to those plays or to Hamlet or parts of Richard II. Even among revenge dramas, however, Titus Andronicus has to take some kind of prize for the gruesome. Horrible things happen to the principal characters, whether they deserve it or not (and as often as not, they don’t), and they keep getting worse.
One can reasonably ask whether there’s anything here other than the sheer atavistic delight in revenge piled on revenge. Different filmmakers have taken a different approach to that question. But I think there is, and it’s worth considering. One ought, however, to approach any of the films with care. Some of them (I’m thinking principally of the Julie Taymor version) will leave you with visual imagery you may be unable to purge, no matter how hard you try.