Not a small subject smolders!
There were certainly hundreds of sieges, perhaps more, small ones, big ones, long ones, short ones, successful and unsuccessful ones too.
1- The attackers turn up, the defenders might try their luck in a pitched battle or skirmish, and if defeated withdraw into the town or castle in question (eg Colchester 1648, Beaumaris 1648).
2- The attackers summon the fortress to surrender, usually by sending a trumpeter with a summons 'In the name of the King' or 'in the name of King and Parliament' as the case may be. A wily defender will try to draw this out as long as possible by querying credentials and inventing reasons to delay responding.
3- Assuming the besieged reject the summons (sometimes they did accept, eg if very heavily outnumbered, with no hope of relief or having received a hefty bribe) then the beisiegers draw up siege lines, often building forts and excavating their own trench lines, run saps towards the enemy and place guns in battery, starting bombardment.
5- This stage might go on for a few hours or some months, with the potential for sallies by the defenders, usually to capture or destroy mortars etc. Occasionally some undermining was tried, various siege engines were used generally unsuccessfully, and occasionally there was a break for negotiation. In some sieges this stage is prolonged until the garrison run out of supplies and surrender.
6- Assuming the defenders haven't given in by now, the storming of the fortress proceeds. Infantry assault the breaches and attempt to scale the walls with ladders, petardiers try to blow the bl---y doors off while not being hoist, defenders sometimes improvised an 'inner wall' behind a breach point. In a town or city, once the way is clear cavalry are often used to rush to the town square and fight the defenders' reserves (eg Selby), at which point resistance usually collapses. The storm is a dangerous time for all concerned and sometimes troops got out of hand and started plundering and killing indiscriminately, but massacres were unusual in England, though common enough in Ireland.
7- Sometimes the besieged retreat to an inner citadel where they attempt to hold out or sue for peace.
The above is a fairly formal procedure, but the ECW saw many cases of improvisation, eg Birch's men infiltrating into Shrewsbury on a dark and frosty night, cunning plans (Royalists in disguise attempting to take Trent Bridge), treachery (Selby, allegedly), and outright luck (a New Model Army cannon shot cutting the drawbridge chain at Bridgewater). Often a garrison might be relieved by a friendly army or re-supplied in a daring exploit (eg Basing House by Gage).
The surrender was usually quite formalised. Both sides would appoint commissioners, usually senior officers, the defenders would insist on lenient terms, the besiegers harsh terms, wrangling would continue until the besiegers lost their patience, often re-starting bombardment to make a point. Quite often (eg Reading) the defenders were permitted to march away to a friendly garrison.
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