Rhythm of Play set out the frame: when timing starts to matter the game tightens into rounds of about six seconds, everyone acts in Initiative order, and a turn buys you movement plus one action. This page is what happens inside that frame.
Everyone rolls Initiative and acts in descending order — the roll is your Dexterity modifier on a d20. (A DM can instead use Initiative scores, skipping the roll: 10 plus your Dexterity modifier.) A creature caught unawares is surprised, which is not a condition and does not cost it a turn: it rolls Initiative with Disadvantage and will probably act late.
Your Speed is a pool of feet you can spend in as many pieces as you like — some before your action, some after, some between two attacks of the same action. Nothing says movement has to happen in one go, or first.
The ground can charge you more than a foot per foot: Difficult Terrain, climbing, swimming and crawling all cost an extra foot for every foot moved. Being Prone leaves you with two options and both are expensive: crawl, or spend half your Speed standing up.
Leaving a hostile creature's reach — 5 feet, unless something says otherwise — while it can see you lets it spend its Reaction on an Opportunity Attack, resolved just before you get away. The Disengage action is how you avoid that, and it costs your entire action to do it.
An attack roll is a D20 Test against the target's Armor Class — match it or beat it and you hit. There are two kinds, a weapon attack and a spell attack, and they differ in what you add to the roll rather than in how they resolve.
You do not need a weapon: an Unarmed Strike is a punch, a kick or a headbutt, and anything you grab and swing that was not built for the job is an improvised weapon.
Two things bend an attack roll harder than anything else on your character sheet. The first is cover — solid matter between you and the target, worth +2 to its AC and Dexterity saves at half cover, +5 at three-quarters, and total immunity from being targeted directly behind total cover. Only the best degree counts.
The second is sight. An attacker its target cannot see rolls with Advantage, and a target the attacker cannot see is attacked with Disadvantage — which is most of what the Invisible condition is actually worth, and most of what the Blinded condition costs.
An Unarmed Strike does not have to hurt anyone. It can grapple instead, or shove — and either way the target must succeed on a Strength or Dexterity saving throw, its choice, against a DC of 8 plus your Strength modifier and Proficiency Bonus, or suffer the effect. A failed Grapple save leaves the target Grappled — Speed 0, and the grappler can drag it around — while a failed Shove save either pushes it 5 feet or knocks it Prone. Grapple alone needs a free hand and a target no more than one size larger than you; Shove needs only the size limit.
And when a melee attack would drop a creature to 0 hit points, you can choose to leave it on 1 and knock it out instead: it falls Unconscious and starts a Short Rest rather than starting to die.
Everything from the damage roll onwards — what the type is for, what halves it, and what happens at 0 hit points — is on Damage and Healing.