So from the perspective of the voters, I don't think much will change. As it is, the United States is fairly evenly split between Democrats, Republicans, and Independents/Unaffiliated voters (note: In this case, it is not the same as the American Independents party. These are strictly people who are registered to vote, but are not registered to vote for a party. They are unable to partake in closed primary elections, which are party exclusive.). To this extent, the I/U vote is quite important, as they tend to swing their vote and currying favor with them, especially in swing states and also tend to follow politics quite closely and examine both sides with some degree of scrutiny (seriously, if you want to feel like the most important person in the world, be a registered independent in a state like Florida... the candidates treat your opinion like it actually matters. I once got one party to stop spam calling me for two weeks by threatening not to vote for them if I heard from them between now and the time I go to the polls).
From a government standpoint, it would most likely first be noticed in the House of Representatives. They are the party of the people and should the third party be popular enough and strong enough, can take a significant chunk of seats in an election year (turnovers tend to happen in the mid-terms, when the President is not up for vote). Since they have a smaller constituency, it is rare for a state to have a solid House Deligation of one party. During the 2016 election, only five states (not including DC) saw a single party majority across all jurisdictions in the Presidential race.
The senate will take longer. If a state becomes a strong state for a full senatorial flip, it will take anywhere for four to six years, depending on how the state is staggard for elections (in a six year period, you will elect both senators once, but never in the same election year, and have one year where niether is up for vote).
Presidentially is where it gets interesting especially from a political standpoint. To become president, you must secure 270 electoral votes (each state gets two electoral votes (one for each senator) plus one for each represntative. It should be noted that electoral votes are not the senator/represtative. With a Third Party, it is likely that neither candidate can reach 270 of 538 electoral votes. If neither candidate recieves this vote, than the House is called into immediate session for what is called a "Contingent Vote" which is initially held among the three candidates with the most electoral votes. Each state deligation gets one vote for a total of 50. To win, the candidate must recieve 26 votes out of 50. Congress will continue voting and deliberating until such an event occurs. The vice-President is chosen in a similar fashion in the senate, with the difference being that it only considers the two leading VPs and all senators recieve a vote (winning requires 51 out of 100).
Having a viable thrid party will likely mean that a vast majority of Presidential administations will be made this way (assuming that the viable third party, hence forth called the Viable Party, pulls equally from both likely Democrat and Republic voters. If their base is more reliably to one party, it's really just changing the name of a party in a two party system). Since the President must be chosen by states voting together, individual delegations must first decide who they will vote for and then there must be enough deligations to achieve a majority... depending on how contentious a fractured trinary system is in congress, this could go on for some time. Should the President-elect and VP-elect not be chosen, congress will chose who will be Acting President until they can come to a choice (traditionally, the VP of the incumbant andministration, as this was once part of the constitution, but is no longer written as a rule... Congress can pick anyone). The Acting President title is rather contentious and only two people in history (George H.W. Bush and Dick Cheyne) have ever officially been called it (John Tyler is often considered but as a VP following the death of the president, he just called himself President and had it. Bush and Cheyne both assumed the role while Regan and George W. Bush (respectively) were sedation for surgery. Notably Bush SR. did not assume the role during Reagan's assasination, despite having the authority to declare it, because he felt it might be seen as a coup... but essentially was the CiC for the duration of Reagan's incapacitation.). Whoever becomes the Acting President will remain so until either a President is properly chosen OR a Vice President is properly chosen, who will assume the Acting President role until the President proper is chosen.
Back to the people, this may not be an experience the country will enjoy and may be a time of unrest and calls for changes to how government is run to come into play. Little of merit will get done, as Congress must choose head of states and if the affair goes on for month, important legislation may be put aside. Other countries may see this as a moment of weakness and sieze upon diplomatic opertunities to fill the void of a stalled United States.
On the benefit side, with a third candidate from the Viable Party, the political rhetoric of the country may be tempored as Democrats and Republicans must cort indpendent-leaning voters away from two rival parties instead of one. States where one party is dominant over the other may find a bigger chance to flip if the Viable party appeals to moderates of the dominant party.
To make voting a bit more likely to favor a third party candidate, it might be changed to Instant Run Off votes (currently Maine is the only state that has them at national level). Essentially if Candidates A, B, and C are on the ticket, then the voter will order them in order of preference (i.e. I vote B as my first choice, C as my second choice, and A as my last choice). Should no majority occur, the candidate with the least votes will be eliminated and all votes for him will be distributed to the second choice of each ballot (assuming B is the weakes, my vote will go to C. If you ordered differently, than A will recieve your vote... assuming C was your first pick). For more than three candidates, leaving blank one candidate will mean he will never recieve your vote (The idea is you're more likely to vote for a third party if neither of the two mainstream parties curried your favor. If the Democrat and Republican party candidates are both pretty unlikable, voting for a third party might be more viable if you could still give it to a candidate if the TPC didn't perform well... might have changed some outcomes.).