The Dolomites have to be one of the most beautiful mountain ranges in the world. There are countless long routes at all levels of difficulty. The rock is limestone, the quality of which can vary dramatically! Many of the climbs have short approaches, and the longer ones can often be accessed via lifts. Definitely a classic and historical climbing destination. Most of the routes are "old school", so feel out the ratings.
Getting There
The Dolomites are located in Northern Italy. Fly into one of the larger cities in the area, or, drive from one of the more major cities to the south.
The Dolomites are a stunningly beautiful range. I'm not aware of another destination that offers so many long routes for rock climbers, without the complications of glacier travel. If you like long, free routes at grades that mortals can climb, this is the place for you. It's also worth noting that the local food is fantastic.
There are two drawbacks to consider. First, the rock can be tricky to protect, especially if you're an American climber raised on clean granite or sandstone cracks. There are crack systems here, but you'll also encounter face climbing with funky pods and holes... consider bringing some old Lowe tricams (the passive kind) to place in the pockets. Locals also master the art of threading holes with slings, though my granite-trained eyes rarely spot these placements. Bottom line: Don't launch off on routes near your max until you have a feel for protecting the rock.
Second, there is a bit of loose rock around. Ask locals about the rock quality on your intended routes. When it's really hot, rockfall increases significantly. Note also that the worst rock quality rating given by many Italian guidebooks is "buon", or good. The 4th Sella Tower, for example, is rated buon, but the top is total kitty litter. Wear a helmet unless you're suicidal.
All that said, you can have an absolute blast cruising these beautiful long routes over postcard landscapes. Choose your routes wisely and enjoy!