'Not everyone has an airport and the Arctic Circle in their garden,' said Gagga, my landlady, Grimsey's Lady Gaga, full name Ragnhildur Hjaltadottir, much prefers her nickname.
You can fly in, (30 mins) walk round the island in an hour or two and fly back to the mainland, or else take the ferry (3 hours), then return later in the day...which is what most tourists do. A bit of birdwatching, quick visit to the craft shop/cafe (which only opens when boat or plane arrive) and take a selfie at the signpost marking the Arctic Circle boundary line (or thereabouts. It's on the move.)
If you want to stay a few days, there's the possibility of a boat trip around the island, once the fishermen have returned to harbour, and weather dependent of course. I was lucky enough to be taken round by one of the fishermen plus the delightful company of Halla, as commentator (also one of the guest house landladies.)
My third landlady, Ummur, is also the cook at the only Restaurant/Pub, 'The Krian' (what else?) and I recommend her gourmet-standard cooking. Interesting to see how the babies and children were welcome in The Krian - at least till 9pm - women rule in Iceland. Saturday night the whole village seemed to be there - teenagers on mopeds revving up and down the one road. (So it's not always quiet!) The men arrived on quadbikes (only a few 100 yards drive from their homes.)
It's clear from chatting how much the islanders love their island. In fact, all my landladies were warm and friendly and I was completely spoilt. The Basar is remote, on the north end and involves the scary walk through the arctic terns but if you can't face that, there's another guest house in the village further south. Or go after August when the terns and puffins have left.