Durbuy has only one frequent busservice running to the outside world. Best is to go there with your own transport and parking is often guided rather good. In the weekends Durbuy is car-free and so then you should reckon with no passing through the valley as well. An own car is usefull if you want to visit the overlook point on the top of the hill just East of Durbuy. Only by car one can make it there easily.
There is a possibility to be driven through the alleys of Durbuy by horse and wagon, but ... Durbuy is so small that it easily can be done completely on foot (and I mean all streets and corners). The town's roads however are everywhere done by "kinderkopjes" (rounded stones laying in a pattern) and so do not think it is wise to walk in high heels here.
Regarding going around Durbuy, it's just very easy to walk in a few hours, since it's quite small.
However the problem is actually reaching Durbuy. First of all: it is important, when doing Internet research to prepare your travel there, to distinguish between the town of Durbuy and the Durbuy district. The latter consists of Durbuy and some other towns.
Now it might be that the tourist information office was quite unhelpful and did not give us good alternatives, but unfortunately, at least in October, the town of Durbuy is very difficult to access. The only tip I found on a Durby Local Syndicate Iniative Website was that the closest train stations are Bomal and Barvaux (around 2hrs from Brussels). Bomal is 7km and Barvaux 5km away from Durbuy town. We arrived at 10am in Bomal on a Saturday and at the Info.Office we were informed that there were no buses going to Durbuy, so we had to walk all the 7km.
However, there seem to be some sporadic Barvaux-Durbuy-Barvaus TEC buses, but in the low season they might not be so frequent. Thus, a good verification of these times would certainly help. Otheriwise, Durbuy is reachable in about 90mins good-pace walking from Barvaux.