Long distance LER buses serve some areas of the Luberon. Apt is the main bus hub. The most useful route is Line 22, which travels from Avignon to Cavaillon, stops in Apt, and then continues on to Forcalquier and Digne-les-Bains.
To get north to Carpentras or south to Aix-en-Provence, you need to change to Line 17 in Cavaillon.
Several LER lines run from Marseille and Aix-en-Provence to Digne-les-Bains; to get to the Luberon, you need to change to Line 22 in La Brillane, or you can ride all the way to Digne-les-Bains and backtrack from there.
The Luberon valley is no longer on any rail routes. The nearest train station is on the western side of the valley in Cavaillon, which is served by regular regional trains from Avignon. From Cavaillon, several LER buses a day run east to Apt.