A church from the eleventh century. There are interesting paintings on the walls inside it, people said to me, but since it's open for visitors only in the summer, I never saw them.
Make a stop in Gränna, the village from which the ferry to Visingsö takes off. It's a very cute little village with smal wooden houses and narrow streets. It is well known in Sweden for a kind of sweets called polkagrisar, which are made there. You can look at the cooking of the sweets and there are a big number of shops that are selling them, plus a couple of cafés and restaurants, (but not that much other things). There is also an interesting museum, the André muséet, where you can learn about the André-expedition to the Arctic pole in the end of the 19th century, and about local history.
This is a church built in the 17th century, during count Per Brahes reign on the island. Originally, however, at the same place was standing a medieval church, built in the 12th century, called Ströja church, and the tower belongs to it. The church is filled with beautiful art and sculptures from that time and ahead. The clock tower, standing nearby, is also beautiful and of a special style. There are more photos from inside the church in my travelogue "Art in the church Brahekyrkan".
A little place where you can eat Swedish cookies, bread and butter etc. and have a cup of coffe, tea, hot chocolate. The owners have collected a lot of things from the beginning and middel of the twenteenth century that they exhibit in the café, for example some old beautiful concertinas. You can also borrow Swedish magasines from the 40's, 50's and 60's to read while drinking your coffe.
...but only during the summer. Since the island is something of a tourist spot things like that really shifts along with the seasons, and when the tourists begin to come in May restaurants and clubs open. And in September they close, one by one...
The titel of this tip is perfectly true if you haven't got a privat boat in the lake Vättern. The ferries leave from the harbour in the village of Gränna, which is an hour by bus or 45 min by car from Jönköping. I'll soon put the e-mail address and an English explanation of the ferries' timetable here. In winter the ferries leave about every hour during daytime, and a bit less often in the evening. In the summer much more often. You can bring your car with the ferry, but I think that in the summer it's probably much more nice to go by bike or horse on the island.
I haven't tried it myself, just seen the wagons passing. But it seems to be a very nice and relaxing way to go around on the island. Possible only in the summer.