Visby is on the island of Gotland, so you are arriving either by air or by ferry. Both options are straightforward from Stockholm, and the ferry is the more enjoyable choice if you have the time.
By air: SAS and BRA fly Stockholm Arlanda to Visby Airport in about 45 minutes. There are also seasonal connections from other Swedish cities. The airport is 3 kilometres northeast of the city centre.
By ferry: Destination Gotland operates car ferries from Nynäshamn (90 km south of Stockholm) and from Oskarshamn (on the mainland opposite Gotland). The Nynäshamn crossing takes approximately three hours on the fast ferry or five hours on the overnight ferry. Book well ahead in summer — the boats fill up with Swedes bringing their cars to the island.