Sangiovese is a good place to start.
Sangiovese is both the most popular type of grape in Italy and the name of a number of red wines that accompany pasta dishes extremely well. This versatility makes it difficult, in fact, to recommend a particular Sangiovese over another, but there are any number of decent recommendations at Serious Eats and Wine Folly.
For our Italian supper, we enjoyed a bottle of Santa Cristina Toscana from Antinori, a fairly accessible bottle found in many parts of the US. It's a good, economical place to start before exploring more of this tasty grape and wine variety.
In fact, you might have already tried Sangiovese if you have enjoyed a glass or bottle of Chianti at an Italian restaurant. It turns out that the most famous Italian wine is required to contain at least 80% Sangiovese grapes grown in the Chianti region in Tuscany in order to qualify as a Chianti. This makes Chianti the very embodiment and flag bearer for the Sangiovese grape, even if most consumers don't know that it's this grape that carries the wine. Might as well add Chianti Classico or Chianti Superiore (Wine Spectator explains the difference) to your list as well.
For that matter, keep an eye out for Italian wines in general and their wide variety of grapes and wines beyond Sangiovese; there is quite a bit worth sampling. From the Nebbiolo grape in the North (look for the majestic Barolo that pairs great with a flavorful steak) to the Primitivo grape in the South (produces the wine by the same name, Primitivo, that complements grilled fare very well), Italian wines are begging for further discovery.
I'll cover more as we explore more wines and regions of Italy. Until then, cheers and cin cin!