The European Union and the Council of Europe are not the same thing, although there are blurred lines between them and patrons researching European legal issues will probably look into both. This is particularly true in the area of human rights, since the European Court of Human Rights is actually a CoE institution, not an EU one.
For a helpful guide to disambiguation, check the CoE's Do Not Get Confused page.
Globalex and ASIL both have excellent guides to researching these organizations and working with their material.
The Council of Europe predates the EU and is twice as large, including observing members like the US. Unlike the EU, it's not a supra-national organization with lawmaking authority over its member states, though CoE membership is the commonly accepted precursor to EU membership. It's mostly a policy-setting group for European countries and their geographic/ideological neighbors, but it does have influence as a treaty-coordinating body and as a player in human rights through its policy efforts and the decisions of the European Court of Human Rights.
1. Documents of the legislative branch, the Committee of Ministers and Parliamentary Assembly, are readily available on their respective websites.
2. The many CoE-sponsored treaties are available with helpful explanatory statements and status information on the page of its Treaty Office.
3. CoE produces voluminous quantities of surveys, reports and white papers, which can be quite useful for researchers. Since they charge for most of their publications, the best way to access them is mostly in print, and Biddle gets a good quantity of these. Some will be available for free on the CoE website, e.g. on the pages of its Commissioner for Human Rights.
4. European Court of Human Rights decisions are available in Westlaw, Lexis, Justis and Oxford Reports on International Law (and keyword searching is in fact easier there), but source hunters should be pointed to its website and its HUDOC database for PDFs. The one frequent point of confusion is whether the PDF a student pulled off of HUDOC is the "official" one for Bluebook purposes. If the case has been published in the official reporter, Reports of Judgments and Decisions, the Bluebook requires they use that version, so point the student to the website for 2007-present and our print volumes for earlier ones. If the case hasn't been published in Reports of Judgments yet, the student can use the HUDOC version. You can tell if the source hunter's version is from HUDOC because it will be in slip format lacking consecutive pagination.