You should rent a bike and bike around the island. You will find many nice roads and places where you can be alone and no other tourists will be.
Equipment: Just a bike!!!! you shall of course have a tent and probably clothes for camping. Why not...
Gotland is known for Sweden's only wild horse, the Gotlandsruss or simply "russ". In English, it is called the Gotland (or sometimes Viking) Pony and it runs wild at the Lojsta Heath in the inland south of the island. The closest I can come to describe them is if you are familiar with Exmoor ponies, as they are similar in size and colour. Personally, I find them quite stubborn but they are ideal for children. Here on Gotland they are also used for pony trotting competitions with a cart. This is how you can aquaint yourself with them if you are too big to ride one (most take no more than 50 kg riders). Gotland is also full of other horses more suitable for adults. Have a look at the first link below for trekking and western riding for instance.
If you are interested in the annual gathering of the wild ponies in July, have a look at the contact details in the second link below. Otherwise, try to spot them by driving to Lojsta any day and look for the signs towards "russpark". Don't expect them to be easy to spot all days though.
Gotland has a bunch of games not found elsewhere in Sweden, even if the old game of kubb has spread to the mainland, but only as late as in the 1990s. You can try the Gotland version of tossing the caber ("stångstötning") as well as plenty of others for instance at Lärbro. If watching the games instead of participating is more your thing, then head for Gotland in July to see the Stånga Games, Gotland's own olympics! See the site below for dates and statistics :)
If you have spent a long time in Scania (Skåne) or Halland you are not that impressed by the Gotland beaches, but the Stockholmers are, and they are still amongst the best in Sweden. Tofta Strand south of Visby and Sudersand on Fårö (see page) are the most famous beaches. The first one also for its party life along with Snäck beach just north of Visby. Sudersand is more known for being a long lovely beach. Gotlanders themselves think the beaches on the eastern coast are the nicest, such as at Ljugarn. If you want to windsurf or similar, Tofta Beach seems popular, being on the windy west coast.