One possible way to visit mount Vesuvius from Sorrento is first by Circumvesuviana train to Pompei and from there with the EAV bus to the top of the volcano. In Pompei the EAV bus leaves in front of the Circumvesuviana train station, opposite the entrance of the archaeological site. The bus trip takes just under one hour and costs € 10 per person (return trip). You can check the bus timetable in advance.
TIP: Another option from Pompei are the trips offered by Busvia del Vesuvio. This company offers a free shuttle bus from Pompei (Circumvesuviana station) to the entrance of the Vesuvius National Park. There you have to buy a ticket for € 20 per person (return trip). It continues with a safari truck to the crater. The journey is spectacular.