If you know nothing about Sivir read another guide.

I always buy the items in different orders because no two games are the same.

Dont read guides to tell you how to play champions, figure out how you like to play them. Reading guides sometimes is counter productive because you are trying to adopt the play style of the person you are reading. Read them to look at builds ignore the other stuff it will just mess you up.

Let me clear up the controversy...

Black Cleaver is good because with her ricochet she debuffs the armor of the entire enemy team.

Bloodthirster is good to buy early on because Sivir can always keep it stacked.

Trinity Force is a bad item on her because her CD's are too long to make good use of spellblade, the AP is wasted, the speed buff after each hit along with her passive puts her over the movespeed soft cap and is not effective, it is not gold efficient for the DPS it gives her.

Ionian Boots and Merc Treads are very viable to replace Berserker's Greaves.

Manamune/Muramana sucks on sivir because you want straight up AD to scale on her abilities.

Infinity edge is a no brainer on an ADC. I am too lazy to show you the math.

Buy whatever defensive Item you need late game. QSS is good sometimes, but sometimes you might want a Vail, Warmogs, or GA.