There are two options when flying to Verbier. The closest airport is at Sion which is 53Km away and approximately 1 hour transfer time although there are not many scheduled arrivals there. Most non-commercial planes can land there. Or there is Geneva International airport, which is a bigger and busier airport 160km away with approximately 2 hours transfer time.
For the road transfer here are a few reputable transfer companies you can choose from:-
For those curious about the prestige of the Swiss railway, the nearest main railway station is in Martigny, which is 28KM and a 40min transfer from Verbier. Alternatively you can catch the local train from Martigny to La Châble train station on the valley floor. From there you can either be picked up by a transfer or catch the telecabine straight up to Verbier itself.