Even if you don't have access to subscription legal databases like LexisNexis, Westlaw, or Fastcase, there's still quite a lot of legal source material available on the internet that you can access.
If you're looking for federal case law (Supreme Court and federal circuit courts), I recommend Altlaw and Precydent (in that order). Justia also has a nice Supreme Court collection.
For federal statutes, regulations, and rules, the best choice is Cornell's Legal Information Institute.
State statutes and cases are much more difficult to research, at least in any comprehensive way. Justia and Cornell's LII act as decent portals from which to begin your search for these materials.
Your online research will be greatly enhanced by Jureeka. Jureeka adds hyperlinks to different types of legal citations and allows you to jump directly the cited source material (much of which is on the sites mentioned above), creating a Lexis-like experience (without the subscription fees).