Yes, it's messy, but I find this Sivir build works. Thanks to one of my friends for giving me a basic idea of what I should build.

Boots allows you to safely farm at the start, I find it the best item to get while starting off. It also lets you get 3 Health Potions, letting you stay in lane a long time (as long as you can actually block the enemy's attacks). Berserker's Greaves lets you fling fast. If you get the chance, you can destroy a turret by over half at level 6 with half a minion wave and boots~
Tear of the Goddess for the Mana, then Manamune for increasing damage throughout the whole game. This is like, the best item ever for Sivir. Though you might disagree, it's never failed me. </3 Also, your enemies will learn about your shield eventually, they won't supply you with mana forever.
Recurve Bow so you can farm easier.
My opinion of Bloodthirster has completely changed recently. I've found that getting one helps boost my damage a lot, and helps me survive teamfights. Also, since you're supposed to be pushing lanes, and you have a Guardian Angel, you should be fine.
Guardian Angel. I cannot stress this enough. There's nothing more fun than suiciding to kill an inhibitor, then Ghost+Flash+Ult your way out of a whole group of enemies. This'll also make you kind of pointless to target first in team fights.
Madred's Bloodrazor isn't my favorite item, but with my Sivir items, I have so much attack speed it works out great. If you're facing anyone tanky, you'll need this.
I've tried a second Bloodthirster, but it doesn't help me keep alive. Black Cleaver seems to help me put out almost as much damage, but help kill people either for protecting myself or for feeding myself/my team. With a second BT you can't do damage to anyone who has a little armor.

Get Shield, then Boomerang, then the sequence is R-E-Q-W.

Now you HAVE to admit, I carried that game. :<
... Or not, whatever.