The main point of this build is that my first item is Philosopher's Stone. Why? HP MP regen early game is something almost any hero benefits from greatly, then you get the money coming in for a jump start on your AD items.

The rest of the item build is focused on the principle of getting obvious Sivir items, but doing so in pieces that are cheaper than many other builds demonstrate i.e. get the slowing effect of Phage earlier, but don't focus on the Frozen Mallet until later game... Bloodthirster is clearly more important, but is also more expensive.

Most of the rest of it is all very flexible in my mind. I happen to like Ignite because of its extra long range. For the same reason, I also like the Gunblade.

In the end I try to play Sivir as an all around team player. She can farm and push well, support with AD and Ult (and Ignite & Gunblade) in a team fight, or port back to base to hold off a creep attack or hug a tower and use Ricochet and Boomerang to harass an incoming team push.

PS - Remember to keep your finger on the E key anytime you're alone. I've saved myself from being sniped by Caitlyn on more than one occasion with Spell Shield.

I love the Rejuvination Bead and Faerie Charm combo to start with. Building into the Philosopher's stone is a great way to keep the cash flowing strong all game long.

With Berserker's Greaves, I think you could get the Boots of Speed, then Vampiric Scepter, before finishing the Berserker's Greaves.

Phage is cheap enough that I really like to get it earlier on.

Bloodthirster and Stark's Fervor are interchangeable. Go ahead and build them one-piece-here and one-piece-there.

Frozen Mallet, Hextech Gunblade, and Infinity Edge are all finishing touches. Any order or preference will do.

Just a few notes here.

Boomerang is your best harassment spell because of its range and damage. You've got to skill that up at every opportunity.

I take Ricochet 2nd because I've go the Faerie Charm from the start and can start farming earlier without OOM.

As I mentioned earlier: keep your Spell Shield on a hair trigger. Anytime danger shows up you tick that spell off immediately and you can often save your skin. I also use my Ult as an escape as needed. I'd rather "waste" my Ult than to die.