A general description of the Swedish Judiciary is available in English at the website of the National Courts Administration (Sveriges Domstolar). However, there are no official translations of cases from Swedish courts.
General Courts: Supreme Court and the Courts of Appeal
The general courts hear both criminal and civil cases. The Supreme Court (Högsta Domstolen) is located in Stockholm. On its website, it provides guiding decisions in chronological order from 2012-present (Swedish only). Additionally, guiding rulings from the Supreme Court can be searched here (Swedish only). Coverage is from the early-1980s to present. In print, select cases from the Supreme Court are published in Nytt juridiskt arkiv. These are located in the law library on the 4th floor in section KKV9 .N98.
The general courts hear both criminal and civil cases. There are 6 Courts of Appeals (Hovrätterna). Guiding decisions from the Appeals Courts can be searched here (Swedish only). Coverage is from the mid-1990s to present. In print, select decisions by the Courts of Appeals are found in Rättsfall från Hovrätterna, which is located on the 4th floor of the law library in section: KKV20.R38x.
Administrative Courts: Supreme Administrative Court & the Administrative Courts of Appeal
There is one Supreme Administrative Court (Högsta Förvaltningsdomstolen, formerly Regeringsrätten). Guiding decisions from the Supreme Administrative Court can be searched here (Swedish only). Select case law is published in Regeringsrättens årsbok, which is located on the 4th floor of the Law Library in the section KKV2764.A57 R44x.
There are 4 administrative courts of appeal (Kammarrätt). Guiding decisions from the administrative courts of appeal can be searched here (Swedish only). There are no longer any official reports for the administrative courts of appeal, but the Law Library maintains print copies of the former source of reports for the administrative courts of appeal: Kammarrättens Årsbok. These are found on the 4th floor of the library in section KKV21.K36x.
Special Courts: the Labour Court, the Market Court, and the Court of Patent Appeals
In addition to the general and administrative courts, there are 3 special courts: (1) the Labour Court (Arbetsdomstolen); (2) the Market Court (Marknadsdomstolen); and (3) the Court of Patent Appeals (Patentbesvärsrätten). Guiding decisions from all three special courts can be searched here (Swedish only).
In print, reports from the Labour Court are found in Arbetsdomstolens Domar, located on the 4th floor of the Law Library in section KKV23.A73x. The Law Library does not maintain reports of the other two special courts.
For translations of Swedish cases into English, your best bet is trying to search the Law Library's case law databases below.