The rules for a fight are on Combat, and this page does not repeat them. What follows is the other half: what you do behind the screen while those rules run.
Roll Initiative once for a group of identical monsters rather than once each — they act together, and you have one fewer number to track. Write the order down somewhere you can see it without asking the table to repeat itself. Then, every turn, say two names: whose turn it is now, and whose is next. That second name is the single biggest thing you can do for a combat's pace — the next player starts thinking about their turn while the current one is still happening, instead of after.
A stat block is something you operate, not something you read cover to cover at the table. Before the fight, look at three things and stop. Armor Class and hit points tell you how long the monster lasts. Speed and senses tell you how it moves and what it notices. Its one or two interesting actions are the only part worth having actually read — everything else you can find in the moment, if you ever need to.
The full monster list carries every monster in the database, with its CR.
Decide what a monster wants before the fight starts, not during its turn. A wolf wants to isolate the weakest target; a guard wants to keep the party away from the door behind it; a cornered thing wants out. With that decided in advance, each turn is a short question — what does it do about what just happened — rather than an open one you are improvising from a blank page.
Play the monster as smart as it is, not as smart as you are. A giant rat does not target the wizard because you know she has fewer hit points; it bites whoever is closest. Do not optimise on the monster's behalf, and do not go back to the stat block mid-turn — whatever you did not already know when the turn started, the monster did not either.
Bloodied is the cue the rules hand you for when a monster's behaviour should change. A pack that was committed at full health might break once half of it is down; a beast that was merely hungry might decide the meal is not worth this anymore.
Cover and difficult terrain are not just difficulty you set before initiative — they are two things you keep placing and using once the fight is running, same as any monster's actions. A monster that ducks behind the pillar it started next to, or drags the fight into the rubble it knows is there, is doing more for the encounter than another point of damage would. The rules for both are on Combat; this is just a reminder that they do not stop mattering once the fight begins.
Most fights should end before the last monster drops. Nothing says a combat has to run to the last hit point on either side. A pack that has lost its nerve flees, a captain who is clearly beaten surrenders, a guard who was only ever doing a job asks to stop — give monsters a reason to want out, and plenty of fights end themselves.
The same restraint runs the other way. When a melee attack would drop a creature to 0 hit points, players can choose knocking it out instead of killing it outright — an option they forget they have as often as not, and worth reminding them of before the blow lands. For what actually happens to a character who drops, see Damage and Healing.