Gothenburg is the 2nd largest city in Sweden. Home to over 500,000 people, Gothenburg is home to the largest university in all of the Nordic countries, the Gothenburg University (over 60,000 students). It is a very old city, dating back as far as the 17th century, when it was founded by the King Gustav II Adolf.
Gothenburg has its own airport called Landvetter. Other common ways to get to the city are bus or train, and there are ships (who have passage to Denmark, Germany, Belgium, and The United Kingdom). Gothenburg's public transportation system is efficient and includes buses, trams, and ferries. Visit http://www.vasttrafik.se/en/ for details on this.
Hostels in Gothenburg (as in most of Sweden) are generally very high quality and the prices are reasonable. Dorm beds will cost you about 20€ a night, and double / twin rooms a bit more per person. Remember to ask Gothenburg hostel staff for tips on what to do during your stay in the city and what's going on during the time you'll spend there.
When you stay at a Gothenburg hostel, be sure to visit the variety of museums, historical buildings, and sites (the Gothenburg Cathedral and the Christinae Church are probably the most famous ones) in the city. Gothenburg is a city with a very high historical value. There are also several lovely parks and gardens. Since there is plenty to do and see, be sure to reserve at least 2 days for the city if you want to do it comprehensively. There are also shopping centers and a nice street for dining called Linnégatan, which is highly recommended.