A sequential list of bills is available in the publication "Digest of Public General Bills and Resolutions"; a digitized version of the Digest is available on HathiTrust.org back to 1936.
Full text of all bills published since 1993 are found on govinfo.gov. If doing a subject search, select "Congressional Record" under the Collection category. The Library of Congress American Memory website has 19th century bills, resolutions, and Senate joint resolutions.
The Libraries have those Congressional publications issued by the Congressional Information Service (CIS) for 1976-1986 on microfiche in the Government Documents area. The fiche set Includes committee hearings, reports, special publications, executive reports and public laws. CIS Annual (Ref Z 1223 .A252x), held in Dewey Reference and GovDoc Storage for 1970-2005, indexes Congressional publications.