The process for nominating and selecting a new Justice of the US Supreme Court is explained in the 63-page February 19, 2010, Congressional Research Service Report entitled Supreme Court Appointment Process: Roles of the President, Judiciary Committee, and Senate.
The President’s nomination role is explained in the Octobeer 19, 2015, CRS Report entitled Supreme Court Appointment Process: President’s Selection of a Nominee.
The specific role and processes of the Senate Judiciary Committee are described in the October 19, 2015, CRS Report entitled Supreme Court Appointment Process: Consideration by the Senate Judiciary Committee.
The US Senate’s Advice and Consent role and processes are described in the October 19, 2015, CRS Report entitled Supreme Court Appointment Process: Senate Debate and Confirmation Vote.