You can get judicial opinions from printed reporters, from Bar Association and Law School Libraries, from comprehensive commercial databases such as Lexis and Westlaw, from less expensive (but not comprehensive) subscription services such as LOIS, Fastcase (1950-present) and Versuslaw (1950-present), from Google Scholar (1950-present, free), from the Casemaker consortium and from free government sites, such as those available through FindLaw.
Bottom line: Use whichever format seems best for the job at hand considering (a) time (b) cost and (c) quality (of both the printed copy and the service).
Multi-State Searching: The fee-based and subscription services all let you search cases from multiple states at the same time. Google Scholar is free, but it still lets you search through any combination of states back to about 1950.
Abbreviations/Citations: To find out the meaning of abbreviated case reporter citations -- or to get the proper abbreviations -- use The Bluebook: A Uniform System of Citation, the relevant volume of Bieber's Dictionary of Legal Abbreviations (William S. Hein & Co.) or the Cardiff Index to Legal Abbreviations. To find out the name of the relevant reporter for a given court at a given time in the past, check out the Bibliographical Index to the State Reports Prior to the National Reporter System.
To get more information about the various courts and case reporters, see Fundamentals of Legal Research (West) and/or visit the court's Web site.
For more information about a specific state, see the entry for the state in question.