The Constitution grants the power to determine voting law to the individual states, so long as they are "republic in nature" (The Founders used republic to denote a system without a monarchy head of state AND using what we would today call a representative democracy).
Within those bounds, states have broad powers as to their voting laws and violations of voting laws are largely handled by investigatory bodies of that particular state (usually the state's "Secretary of State", which is more of a department of the interior, rather than the federal branch's which is charged with foreign affairs).
Beyond the requirements for a constitution denoting the relationship of the three branches, you can have Westminster styled republican parliament, a Swiss-style semi-direct democracy (all U.S. states are in fact semi-direct democracies), a unicameral non-partisan legislature (one state does in fact have this) and your elections can be instant-runoff (Maine now has instant-runoff elections for all offices).
Weirder still, there are no rules to determine how to assign elector votes in the Electoral College, and in theory, "The Governor tosses a coin, heads the incumbent party candidate gets it, tails the opposition party gets it. On its side, with head facing away from the ground, Libertarians get it if not the incumbent party, unless the head sees its shadow, then for six weeks of winter, the Green Party holds office instead of the incumbent" are perfectly acceptable guidelines for delegating those (though 48 states are winner take all, with two states designating one to each congressional district and the remaining two are given to the statewide plurality winner.).
So to get a complete shift off first past the post, each of the 50 states would need to independently make the switch to another vote counting metric.
To outright abolish the Electoral College, it would take a constitutional amendment, which requires 2/3rds of both houses of Congress and 3/4 of the states (via passing in each state legislature, 49 of which require two houses to vote in some majority. Good luck with this as the Electoral College is a check against mob rule) OR by Constitutional Convention, and no one really knows how those are supposed to work and no one really wants to either (mostly because no one is sure if you can set one to discuss specific amendments or if you are required to allow any amendments from any delegation to be discussed... which is essentially a nuclear option no one wants to press the button on yet. The only thing that couldn't get rewritten is that each state gets two senators).
The above system would also be required if one wants to move the power to change election counting methods from a state power to a federal power... and then you need to get Congress to agree on the new standardized methods.
