When I started playing Sivir was in the free rotation, and randomly selecting a champion I went with her... Instant love after the first game.

Then I started reading and learning about the game, and from the various posts in forums that I read, I uderstood that Sivir was not that good a champion, that I shold go with others since She was Nerfed and stuff... (I didn't play her before, but if this is Light Sivir, OMG I really wish I could play with the old one!)
Played various champions and went back to my old love, and then built her properly, at least for MY playstile...

Just max out Boomerang, take on the hunt when available, and then get one early Ricochet and one Spell shield, from then on, read the opponents build. I max out Spellshield against a Ryze or Annie, but if you are against mostly melees go for the ricochet's, but remember that spell shield is OP, you can watch Warwick scratching air as you move out, Fiddlesticks with his arms up thinking "why am I not sucking life?", a well aimed Ashe Arrow that does not do a thing... every champ gets a WTF moment when you time yours Spell Shields right, just remember, it doesent work on AoE spells.

I'm still playing with this since I came to LVL20 a few days back, but essentially I want to
1) Regen my mana: a lot of Vlads scaped for I had no mana for that last boomerang early game
2) Attack faster: I don't know what people mean by true DPS but attacking 112 x 2.5 times a second with doing 90+% crits is fine by me
3) Support my boomerang: a bit of AP wont hurt

This matters only Early game, later items will do the rest.

Utility FTW, after that it's your choice, play D, or go all out. I go 9/0/21 but that's really up to you.

Teleport FTW, allways teleport. It's nice to see a green dot against a red swarm, teleport and get instant 200+ gold of exploding minions, and aditionally taking a turret. Or ganking an overly confident tank near your turret.

I use Heal as my other spell, yes heal. "What a Noob" is what people say... well, while you hit me early game, and exaust me, and ignite me, I just keep hitting you, then heal, then kill you trying to run away, hell maybe then I'll even take your turret minute 4. "But Potions Dude, use potions!" Potions costs. I want to buy a Zeal by the time you equip full boots. Every cent counts, so I "sacrifice" one summoner spell to gain a lot. Of course you may play a diferent style, this is what work for ME, never forget that when we meet on the lane just before the nice lady sais "Welcome to Summoner's Rift"

Energy Regen? Mana Regen? Gold? Some will say, it's a newbie thing to worry about gold, well my friend, I don't care, I like to have my items ready when I'm stil lvl 15 and you trying to pay for that warmog so you can start prepairing for late game...
Start with Regrowth Pendant, when you have 375 gold (2-3 minutes of play) go back and purchase Philosopher's Stone. It's important not only to your economy but for your mana regeneration as well. Then teleport back, your opponent will be still traying to kill the last minions you left with him. Now you have a little (very little) mana regen, so you can afford your Ricochet (a little and only to push)
When your teleport refreshes you'll have enough to buy several things. Preference: Maki Pendant + Boots + Tear of the goddess + Null-Magic mantle + Mercury's talons. If you did a few kills and to all the minions you took the final hit, then It'll be enough for all of this.
By now a turret is no longer there, you can choose to push to the Inner turret or gank/support other lanes.
Then it's faily simple Get Manamune, never deactivate ricochet from then on, and start building zeals from Daggers. You need Phantom Dancers, but, never let any gold in store when you recall, so allways buy zeals, or daggers...

Finally, if you still haven't carried your team to victory, then remove the philosopher's for an Infinity edge. You have 99% crit chance, so, it'll hurt quite a lot.

You have to read the game. Sometimes Nashor's Tooth is good, you get a little AP boost, and a cooldown reduction to go with your boomerang. Sometimes you just need to get Defense but if it's going that way, maybe the game is already lost. Stark's Fervor is awesome if you must switch to support. But every choice you take comes from une less Phantom Dancer don't mess with your Philosopher's + Mercury's + Manamune awesome base.

Boomerang Blade when available. Then lvl2 Ricochet, and then if you're against casters (almost always) spell shield, you can see the WTF face when a Ryze tryes to snare you and fails to his death. You can see it when an Ashe times an arrow perfecly that does not stun you... You'll do fine with lvl1 ricochet anyways, so prioritize your spell shield, it rocks.
And ofcourse On the Hunt when available.

I never jungle, but when you're lane changing I like to have my lizard's stew hot and steamy. Don't bother with golem, you 'll have all the mana you can ever need, and then some.

You'll be going to dangerous places alone quite a lot. But when you can, abuse team fights. Be the Rajon Rondo of LoL, you'll have every single assist you can hope for. And when they try to get away? You just Bomerang their ***es...

Abuse her. When your team has selected at leat a Tank, grab her. But I've won games with no tank at all... patience is the key. Teleport to the right place at the right time, blow a turret, push a little more, then dissapear.

When someone request mid, fight for it (Politely). But if you have to go Top or Bottom, go with a melee champ. Focus on one enemy at a time. Your teammate will have exaust or some stun at hand, roll with it. Don't leave him alone. You're not a tank, but with this build you can take some before running, you'll regen it anyways.

2v1? Let them push, turret hug, sit on top of it if you can. None can aproach it, and minions will keep exploding.

You'll be melting minions waves into gold.

Plese, feel free to comment. I'm hoping for some constructive feedback.

Enjoy!