Most backpackers who travel to Sweden just visit Stockholm. They see the city sights, make excessive jokes about the ridiculous prices and then return to a more moderately-priced Europe. Few travellers get off the beaten path in Sweden mainly due to the fact that scandinavian countries are hella expensive. Understandable, but unfortunate, since each of them has a lot to offer beyond their urban centres.
With such as vast wilderness to the north (including the epic Kings Trail) and picturesque coastal towns dotting the west coast I am always surprised that more people don't adventure outside of Stockholm. So, for anyone looking for an off-the-beaten-path trip in Sweden that will bruise the bank but hopefully not break it, I will elaborate.
First: Visit Göteborg
Göteborg (Gothenburg) is a nice stopover between Denmark and Norway. It's smaller than Stockholm, and is a beautiful, quirky city that you can easily explore on foot. It also will connect you to the smaller towns in nearby Tjörn. From Göteborg you can take a bus to Tjörn which will connect you, via ferry, to three quaint little islands you can explore over a few days.
The Island of Marstrand
Marstrand is the most luxurious of the three islands discussed here. It is the largest and most populated, with around 1300 people spilt over its two halves. You can actually reach part of the island by car, though most people access Marstrand by boat. The main reason to visit here is the 17th century fort, Carlsten. It has a very interesting history, and since many signs inside the fort are in English you can read and explore with ease. The fort has some cool underground passageways to navigate and a gorgeous view from its ramparts. On top of that, there are lots of little shops and stalls open throughout the summer as well as a few restaurants AND night clubs, making Marstrand a surprising destination for any wandering backpacker looking for an unconventional destination to explore.
The Island of Dyrön
Dyrön is a great island for hiking. It has a few paths around and across the island that will allow you to explore the rocky coast as well as the semi-forested interior. The view from the peak of Dyrön is a beautiful place to stop and picnic lunch. Better yet, with only 250 people living on the island you can easily camp on the island and enjoy the view until your heart's content. The island has all the necessities you'll need, as well as a quaint little beach. The ferry comes and goes almost every hour from dawn until dusk, so it is quite accessible from the bus stop at Rönnäng and it's only a stone throw away from Åstol.
The Island of Åstol
Åstol is the smallest of the three islands, a mere rock jutting out from the sea in the Skagerrak strait. It's shockingly tiny, though even with 200 people living there it feels cozy, not cramped. It's an almost surreal destination as it has no roads or cars; it's a far cry from the hustle and bustle of any modern city. Yet Åstol boasts a cafe, restaurant, AND small grocery store. It's harbour is full throughout the summer with boats of all shapes and sizes, with people from all across northern Europe. It's a wonderful place to escape for a day, relaxing on the warm stones before taking a dip in the ocean. It sits close to the mainland, and right between Dyrön and Marstrand, making it a good starting or ending point for your island-hopping adventure.
As For Accommodation...
This far off the well-worn backpacking trail you may be hard pressed to find affordable accommodation. You will either just want to visit for a day and pick your top island destination, find a couchsurfing host you can bunk with to allow you to explore longer, or camp. Camping is free and each harbour has facilities which you can use to freshen up. I am a huge fan of couchsurfing and would strongly recommend that option if you don't happen to have a tent, but even just making it out for a day is a worthwhile excursion and will give you a glimpse at an area of Sweden virtually unnoticed by the average backpacker. I promise it will be worth the hassle.