Once introduced and assigned a number, bills are immediately assigned to the Congressional committee and/or subcommittee charged with responsibility for that policy area. Much of the work involved in moving a bill through the legislative process is done in committee:
The committees and subcommittees may hold hearings, revise (or markup) draft bills, and recommend passage (or report the bill out of committee). The vast majority of bills "die" in committee and are not referred to the full House or Senate for consideration.
During hearings committees call on executive branch officials, "experts" in appropriate policy areas, and interest groups from the public and private sector to testify to the merits of proposed legislation. Many (though not all) of the transcripts from hearings are published and available in print, microfiche, or electronic format.
These documents can be particularly useful in tracking the stance of administration officials and the opinions of interest groups as well as understanding the positions of senators and representatives through the questions they pose.
In addition to hearings, you may also find that there are committee documents and prints. These are materials produced for use by the committee during its deliberations and can provide background and context for the issues being discussed.