I'm curious what actually happens after the president is impeached. Does he have to take it upon himself to resign? Do the speaker and the majority leader march into his office, hand him a letter and tell him to get out? Or something else? My question focuses on the actual process of getting him out rather than what happens after and who takes office.
After the President is impeached by the House and convicted by the Senate, the President ceases to be the President. There is no precedent for this actually happening, because in all historical cases, the President has either resigned prior to a conviction in the Senate, or the Senate did not convict.
But, there is ample precedent for a President having to leave office at the end of his Presidential term and those precedents would apply with equal force in the case of an impeachment.
So far, every President whose term has ended has voluntarily relinquished the post in a peaceful exchange of power in accordance with the rule of law, although sometimes with a nasty prank or two thrown in as a parting shot in the White House by the President (or more likely by a disgruntled staff member under the President acting without official authority to do so). It has never been necessary to refer such cases to the courts thus far in U.S. history.
There is a rarely used legal action called quo warranto that provides precedents at the sub-national level and in British history, within the court system, in the extremely rare cases where court intervention has been necessary, and if the need arose, those precedents could be invoked by the federal courts if a President removed from office in the impeachment process refused to leave voluntarily in a prompt and reasonable manner.
It would still be more tense, however, because there is a period of more than two months for a transition to take place following an election, while a transition involving an impeachment would take place more or less instantaneously, and it would not be physically possible for the former President to move out of the White House and to transfer all of the incidents of the Presidency to the new President instantaneously. One would assume that a few reasonable allowances would be made to reflect that situation. For example, in cases where the President has died while in office, his family has been allowed a reasonable amount of time to move out of the White House even though the new President took over authority for doing the job immediately.
One could also image a new incumbent President simply bringing an eviction action in the courts of the District of Columbia to physically remove the outgoing President from the White House, although that wouldn't address the need to assert control over the various government departments and agencies. The Secretary of State's office and the Treasury Department are vested with authority of a lot of those mechanics (e.g. changing over payroll arrangements from the old to the new President).
One reason that you would not expect much difficulty in the case of an impeachment in the U.S. is that the President will usually be turning over power simply to the Vice President who is a pre-designated successor to the President and a member of the same party, who has lived through the events of the departing President's administration. Thus, the mechanics of successor would usually be an uncomfortable but cooperative "domestic" affair within the party and not a fight between enemies.
Does he have to take it upon himself to resign?
No. Once the Senate removes a President from office he can't resign because he no longer holds that office which is now vacant.
Do the speaker and the majority leader march into his office, hand him a letter and tell him to get out? Or something else?
Generally speaking, the President would be present in the chambers of the Senate at the time that the vote is made and so there would be not need to notify him of the Senate's decision. In recent history, during the Clinton impeachment, for example, the President also had attorneys who were representing him present in the Senate chambers who could advise him of the outcome immediately.
The Senate clerk would certify the result and deliver notice to the President in the same manner that it does in the case of laws that are passed by Congress, but ordinarily, it wouldn't be necessary to take any action to enforce that ruling if everyone behaved according to the rule of law. No court or senior governmental official would take seriously a removed President's claim that he didn't know that he had been removed from office for more than a matter of a few hours. Any more elaborate enforcement scenarios would be basically speculative.
If Trump was in the White House at the moment he was convicted by the Senate and was immediately notified of the outcome by his attorneys, I suspect that Pence would be immediately sworn in somewhere in the Senate. At that point, I also suspect that Pence would give Trump 24-48 hours to get his "things" together and DT would be given government transportation, including air flight, to a destination of his choice within the U.S. The rest of his personal effects would then be gathered by government personnel under the guise of someone in his family who knows those personal items he owns. The VP would likely remain at the VP residence until at least a majority of DT's items were removed.
Now for the matter of whether DT would leave of his own accord. If he would not, I suspect an officeholder in one of the three branches of government would ask a federal judge for an eviction warrant so that he could be legally removed from the White House. I further suspect the law enforcement officials who would execute such a legal notice would be U.S. Marshals.
So there you have it. A lay person's version of how the events would unfold. All other suggestions and opinions are welcomed.