Speaker of the House.
Although the Speaker of the House would assume the presidency if both the President and Vice-President die or are rendered incapable of carrying out the duties of the office, to date this has not actually happened except in fiction.
Practically, no. Since the vice president is made president if the president is incapable of performing his duties, they would be constitutionally unable to fulfill the obligation if the president were to resign or take ill.
When the President bites the dust, or is incapable of holding the Office of President Article 25 of the Constitution can be put into effect.
The Speaker of the House would become president if both the president and vice-president were unavailable.
Biden would then nominate someone to be vice-president and if he is confirmed by both houses of Congress he would be the vice-president.
the Speaker of the House
Congress would have to pick a new vice president as they did when Nixon resigned and Gerald Ford took over as president. Nelson Rockefeller was picked to be vice president.
The Speaker becomes president.
The secretary of state
No. It is quite unlikely that this would ever happen. When the Vice President becomes President, a new Vice President is selected and becomes next in line. Only if the new President were to die before a new Vice President was selected would the Speaker of the House become President. Then a new Vice President and a new Speaker would be selected and they would both be ahead of the President Pro Tempore in the order of ascendence. The Secretary of State would become President only if the President, Vice President, Speaker, and President Pro Tempore all died at essentially the same time.
The running opponent from the previous election would take the presidency as with the vice president.
If both the President and the Vice President are unable to serve, then the Speaker of the U.S. House of Representatives would become President.