When Hillary Clinton's nomination for secretary of state was first rumored, I wrote here about the process of replacing her in the Senate.
This article in today's Washington Post describes the current state of play. There has been no significant change since my earlier post.
I assume that Clinton will not resign from the Senate until her Senate colleagues confirm her in her new job. That can't happen for a while.
While some media reports might say that Barack Obama has nominated Clinton, or that he has nominated Tim Geithner, etc., technically he can't nominate anyone yet, because he isn't president yet.
The standard process is for Senate committees to hold hearings on a new president's nominees, shortly after the Senate goes back into session in January. Aside from any nominee(s) that are particularly controversial, the Senate readies itself to confirm the nominations as soon as they are formally made. That takes place at the Capitol, immediately after the inauguration ceremony. The new president signs a paper formally nominating his Cabinet, and the Senate then votes to confirm them.
One historical example of a new president's Cabinet nomination that did not proceed routinely through that process was George H.W. Bush's nomination of John Tower to be secretary of defense, in 1989. Senators balked at confirming that appointment, in part because of allegations that their former colleague Tower had a drinking problem. The confirmation process stretched into March of that year, when the Senate voted 53-47 against confirmation.
The Senate later confirmed Bush's second choice, the House minority whip. In light of recent history, it seems strange to label that nominee noncontroversial, but at that time he was: Dick Cheney. Another historical aside: Cheney's House leadership role was taken over by Newt Gingrich, part of a sequence of events that resulted in Gingrich becoming speaker of the House less than six years later.
So far, there is no indication that Obama's nominees will have such problems. The one who is perhaps most controversial, Larry Summers, will not need to face the Senate confirmation process, because his new role, heading the National Economic Council, is a White House staff job.