The legislative process in the US Congress is not quite as the "traditional how a bill becomes a law diagram" describes. Instead in recent years new practices have developed. Barbara Sinclair has documented these changes in her book "Unorthodox Lawmaking", which is about to come out in its fourth edition.
In the book examples are given of different procedures which have been substituted. There are also two books which describe in detail the process by which particular legislation made it to the statute book.
In 1973 Eric Redman's book "The Dance of Legislation" came out. It is an insider's account of how the National Health Service Bill (S 4106) was drafted and passed. The most recent edition includes a postscript written in 2001 reflecting on the changes inn practice.
A few years later Paul C Light wrote a similar book - this time dealing with the passage of the Department of Veterans Affairs Act in the late 1980s.
One of the best ways of improving understanding of the legislative process in the US Congress is to use the resources of C-SPAN. Details of key bills in Congress can be accessed here. C-SPAN 1 carries the live feed from the House of Representatives while C-SPAN 2 carries the live feed from the Senate. Archive material is available at C-SPAN Video Library.