The easiest ways to travel
Travel by air
The quickest way to travel is of course with airplane. During the summer season, some airlines are offering direct flights to Visby (e.g. Airberlin from Berlin, Germany, or Gotlandsflyg from Helsinki (Finnland) and Malmö). The rest of the year, you have take a plane to one of Stockholm's Airports, and get another Airplane to Visby. If you want to catch a cheap airplane, for example Ryanair, you will go to Stockholm Skavsta. Here you will find some of the Airlines traveling regularly to Stockholm:
Quite regular during the day, Gotlansflyg and Nextjet are offering flights from Bromma and Arlanda to Visby.
Changing airports in Stockholm - Shuttle Bus
It is easy to change airports by using one of the shuttle buses. Just keep in mind that it will take you around 2 to 3 hours to go from one airport to the next one. Flygbussarna is offering shuttle buses between all three airports of Stockholm, as well as to the ferry in Nynäshamn.
Travel by ferry and car
Destination Gotland is offering regular ferries from the mainland to Gotland.
They are departing at least once a day from Nynäshamn and Oskarshamn. You can travel as a single passenger with the ferry, as well as with a car, and of course with a horse trailer, too.