TBH I think most people's lives will go onward as the political battle rages in Washington, but I could see some militia groups causing some issues- i.e. in the 'blue' southern states, or in areas with state separatist sentiments (i.e. Eastern Oregon/Washington, Northern California).
From there onwards, I think punitive actions by state police against those groups may draw in more groups looking for action- much like the beginning of most civil conflicts (police crackdown>seen as proof of government violence>government is now portrayed as occupying force>draws in more extreme groups).
Antifa may probably take this national preoccupation to set aside some autonomous zones for itself in the cities, or depending on how glowy they are, act as a leftist/anarchist militia group. You may get some boogaloo gangs in the city as well, looking for happenings.
Simultaneously, I think that each President will attempt to portray themselves as the legitimate president, and try to win international recognition (some countries like China/Russia/Turkey will take this opportunity to pursue their agendas as the US is distracted). China may also attempt to mess around with the US supply chain, given how dependent the US is on it for many basic goods.
Biden may also have a leg up as the Washington establishment and the DC governmental apparatuses may recognize him first, and may attempt to put the opposing states in a vise by denying services (if the opposing states get through it, it also sets off further divisions as you now have parallel bureaucracies).
Military command may try to stay neutral/side with Biden, and I could see a number of personnel 'defect' to their home states depending on how hot it gets, but I don't it'll ever break out into actual pitched fighting.
But honestly, this is just speculation- rate this autistic, but it's all in the air and honestly sort of unprecedented (yes there was the 1877 Compromise, but I don't think it would fly nowadays).