The Dolomites have no airports of their own. The closest options are Innsbruck (Austria), about 90 minutes from the western valleys; Venice Marco Polo, about two hours from the eastern valleys; and Verona, about two hours from the southern edge. Bergamo (serving low-cost carriers) is a longer option, as is Bologna. Hiring a car at any of these airports makes the most sense for exploring the mountains; public transport into and between the valleys is possible but slow.
By train, Bolzano and Trento are the main gateway cities on the Brenner railway line, connecting Munich to Verona. Both have bus services into the main Dolomite valleys. The Val Gardena bus from Bolzano takes about an hour and a half. Cortina d'Ampezzo is best reached by bus from Dobbiaco (on the Brenner line) or from Venice.
In summer, the Dolomiti Bus network covers most of the valleys; in winter, ski bus services connect the resorts within each ski area. Driving gives you far more flexibility, especially for reaching the smaller passes and the quieter eastern valleys.