Do all the votes in a precinct go to the majority candidate (eg winner take all) and then the tallies added up by precinct? Or do they take the simple majority in the state?
The 17th Amendment to the United States Constitution established the popular election of United States Senators:
The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years; and each Senator shall have one vote. The electors in each state shall have the qualifications requisite for electors of the most numerous branch of the state legislatures.
As the US Senate website explains:
The Seventeenth Amendment restates the first paragraph of Article I, section 3 of the Constitution and provides for the election of senators by replacing the phrase "chosen by the Legislature thereof" with "elected by the people thereof."
The candidate that receives the most votes in the state will win the Senate election.