Sivir, in my opinion, is one of the most under-rated champions in the game. She's very versatile in that she can push a tower and retreat before most people can get to it. She can defend against multiple minion waves like almost nobody else can for very little mana. Unfortunately, she's not too good "mano y... uh womano". One last thing, this is how I play Sivir. Defensive early game, aggressive later. That said, let us begin!
Quick note:Team one is mid-lane Sivir. Team 2 is Non-mid Sivir. Just in case the team names don't feel like showing.

Contrary to most builds, I'm going to start out with the items section first simply because I think it will require quite a bit of explanation.

Starting with a Vampiric Scepter and Ricochet will keep you in lane for a long time, no matter what lane you end up being in.
Stay out in lane as long as possible. If you have the cash, recall and buy Wriggles Lantern and parts for Tear of the Goddess if you can afford them.

After buying your lantern, Teleport back to whence you came and fight on. If you took mid lane, place the ward at whichever river side has the tougher opponents to you. If in one of the side lanes, place it in the bush on the river or at your closest "boss" jungle creature, be it the dragon or Baron Nashor.

Here's where the path splits. if you're solo, you want to get Manamune to keep your mana up as much as possible. Otherwise, get an Emblem of Valour Emblem of Valour for the nice bonus it gives to your lane partner.
Bringing the paths back together, grab whatever boots you feel comfortable with. I usually get Boots of Swiftness so I can get to and run away from combat/turrets if needed. But they can easily be substituted for Mercury's Treads.

Afterwards grab whatever you didn't back at the split and upgrade your Emblem of Valour Emblem of Valour into a Stark's Fervor or work on buying a The Bloodthirster. Then do the other one. After that, work on building some defense. You really need it.

Unlike most squishy champions, you should probably not get a Banshee's Veil if you have good enough timing with your Spell Shield. Plus, your shield is much more reliable. If you don't have good reflexes, just buy one.
Side note: Your Spell shield will block the "blinding" effect of Nocturne's ultimate if you happen to have it up while Banshee's Veil, does not.

If AD people are just trolling you hardcore, buy a Thornmail and troll them right back.

Magic people trolling you? Get a Force of Nature. It gives you some tank-itude to magic and tank-itude in general.

If you're dying too much for The Bloodthirster to be effective, get it to full stacks by jungling and buy yourself a Guardian Angel. T.B.H you probably shouldn't be dying anyway.

If you figure "Herp-derp, I'm too good for defense!" just buy another The Bloodthirster or The Black Cleaver to help in team fights.

But honestly, you do need defensive items.

Passive: Fleet of Foot

One of your main survival skills. Increases your dodge chance by up to %25 while running and gunning. If you stop, you have roughly a quarter-second to keep this bonus.

Ability 1: Boomerang Blade

Your main source of damage output. Unfortunately, I don't like being aggressive and I usually end up with this being low leveled late game. It's great for a semi-guaranteed hit on an enemy through many, many minions and/or walls

Ability 2:Ricochet

Ricochet, how do I love thee? Let me count the ways... 1. Great, but somewhat unreliable, harasser. 2. Best farming tool out there (IMO). 3. Unless I'm sadly mistaken, every hit procs life steal! Hence, Ricochet is good.

Ability 3:Spell Shield

Oh the fun mind games you can play with this. Karthus trying to kill you with his ultimate? Not anymore! You just gained 75 mana! Congratulations! If they have a Warwick who is MIA at the time, it's worth it to level this up and use it before going over to a bush. That suppression tied in with his swings huts bad man.

Ultimate:On The Hunt

Bumps speed, attack speed and general awesome-ness of units around you when you activate it. As far as I know, it's not an aura effect so, just running around with it on and then finding a team fight wont help. Activate it once the team fight has started. Don't be afraid to use it to run away from fights where it's just you and minions. It's cool down is short enough that it should be up most of the time anyway.

Now, if you look up top, you will see that I'm not using the "standard" twenty-one in one tree, nine in another. Now you may be saying to yourself, "Derp, down vote time!" but hear me out. She does not really benefit from having the twenty-first point in either of the trees I invested in.

Both of the mastery tree "paths" have the increase gold box checked because she has some really expensive items. And, even though she has an amazing farm-game, sometimes it just isn't good enough. Especially against people who play a mean Karthus... I hate those people!

Side-lane Sivir has increased buff duration due to the fact that she can jungle buffs semi-easily. I also didn't have anywhere else to put the other two mastery points.

Mid-lane Sivir has increased mana regeneration because she truly runs out of it too fast without a Tear of the Goddess. She also runs out fast if you mis-use your spell shield, spam your Boomerang Blade or, have Ricochet on for lone minions. Don't do that last thing. If you do, I will find you and hit you.

Runes are more of a "Which do I want more of today?" kind of thing. The two rune sets I have up top are interchangeable and should probably stay as they are.
Glyphs for mana regeneration.
Seals for health. Per level because you should not be over-extending or playing aggressively early on.
Marks for armor penetration or attack damage, whichever you prefer.
Quintessences for the same thing as marks, just the other way around.

I take Clarity simply because I find myself running out of mana too much early game and not enough late game. If you find something else, by all means, use it.

Teleport is just to get to wherever you need to. If need be, you can teleport to the ward you hid in the bush and scare the hell out of some would-be ganker. Oh, you didn't use the free ward that Wriggle's Lantern gave you? Huh... Oh well, your loss.

Replacements:
Ghost is a nice replacement for Clarity when you don't feel like popping your ultimate ore Teleport to get somewhere.

Cleanse, again nice replacement for Clarity when CC gets you down.

Heal, for when you don't have mana problems. If you have good reflexes and aren't laning against a "spam-caster" as I call them, you won't have mana problems. Unfortunately, this is hard to predict. Karthus, Malzahar, Annie, Ryze=spam-casters Everyone else with high cool down skills=not spam-casters

Revive, sure you can take this. it won't be amazing but, a dead Sivir is a useless Sivir so, you be the judge of this.

Things not to take:
Everything else

Just turn on Ricochet, throw an auto-attack or two and go make a sandwich. They'be be dead before you come back. Of course, you might want to be there in case you almost die while this is happening.

You are the battle mistress. That doesn't mean you can charge into battle, cross-blade ablaze and always come out un-scathed. That said, wait until you see a team fight start... Then charge in there, Ricochet active, Boomerang blade queued up, On the Hunt ready and, a steady finger near your spell shield button. Happy hunting!