Sivir is a range AD carry champion. she specializes in early game damage and mobility.
Sivir is one of the strongest pushers in the game(once was the strongest).
She was completely changed after the Volibear update and now she has longer range, stronger damage and very high chasing potential.
But was denied of her pushing/farming skills.

Pros

Sivir Has incredible damage in the beginning.
Sivir can hit multiple targets with almost all her spells.
Sivir can block an ability (stop a nuke).
Sivir can chase easily.
Sivir can help her team alot with her ult.
Sivir does very good harassment.
Sivir is a great pusher.

Cons

Sivir is very squishy.
Sivir can lose a game if you delay it to much(low damage late game).

AA = Auto-attack (Basic Attacks)
AD = Attack Damage
AP = Ability Power
AS = Attack Speed
AoE = Area of Effect
B = Going back to base
CC = Crowd Control (fear, Stun, snare, slow, etc)
CDR = Cooldown Reduction
DoT = Damage over Time
FB = First Blood
HP = Health
LS = Life Steal
MD = Magic Damage
MP = Mana
MR = Magic Resistance
MS = Movement Speed
OoM = Out of Mana
Re = back to lane (after SS)
SS = Switch side
SV = Spell Vamp
DPS = Damage Per Second; also: Strong Auto-attacker
MDPS = Magical Damage Per Second; also:Strong spells; casters
MP5 = Mana Regeneration per 5 seconds
HP5 = Health Regeneration per 5 seconds
ArPen = Armor Penetration
MPen = Magic Penetration

Your core build

Doran's Blade - This item give you a good start with some health, damage, and a little bit of life steal.

B. F. Sword - You should get this item as fast as you can. when you get it you will start doing some serious damage.

Bloodthirster - This item is a major item for Sivir and for any other ad carry.
It is the highest damage and life steal in the game.
*This item can be bought 2,3 times.

Berserker's Greaves - This is the best boots for a normal ad carry champion.

Black Cleaver - This item gives you a very good damage as well as some attack speed.
*This item can be bought another time if needed more attack speed.

Situational items (More damage)

Infinity Edge - If you are doing well you should buy this item. he gives you some critical strike and good damage.

Sword of the Occult - if you are doing extremely well you should buy this item (try to buy it as soon as possible, maybe even somewhere in the core).

Last Whisper - If the enemy team bought many armor (careful not to confuse it with health) you should buy this item.

Phantom Dancer - This item gives you Criticals, attack speed and movement speed

Madred's Bloodrazor - If the enemy team bought many health you should buy this item.

Hexdrinker - This item can help you to deal better with a ap enemy.
*This item can really cause trouble to Karthus.

Situational items (Defense)

Frozen Mallet - This item gives you a little health as well as damage. And the best part is that it slows the enemy your chasing.

Guardian Angel - If you feel that you die to quickly this is a very good item to you. It gives you armor, magic resist and the best part is that if you die you can revive yourself in the same spot.
*Try to "e" immediately when you're back

Situational items (Mana regen)

Tiamat - This item helps you in 3 ways:
1) It gives you good AD.
2) Sivir is very mana hungry, this item allows you to stay longer in team fights.
3) This item gives you some health regen that can help to stay in team fights longer and might even help you survive.

Manamune - This item gives you many mana and convert all the mana to AD but it is only good once it is full/charged for ~700 if not don't buy it yet.
*If you are going to buy this item buy Tear of the Goddess as soon as possible, maybe even somewhere in the core.

Philosopher's Stone - This item is good for early - mid game in order to keep you longer in lane / battle. (sell it when but not to early because of its passive)
*This item gives you faster "money regen" so buy it as one of your first items if you are planning to buy it.

Situational items (Support)

Zeke's Harbinger - You can support your team with this item (the ad part of it). as well as some attack speed and life steal.
*This item is better if you got more ad on your team.

Best Spells

- Sivir is a very squishy champion. Because of that she needs a way to escape battles quickly. Flash is the spell that can help you do that.

- This spell can help you and your team to chase an enemy champion and get a kill on him.

- This spell is a great spell in dominion game mode but I am not so sure about the classic game mod.

- This spell is very good for Sivir if you are in the mid. It can save your turrets and it can save you from being under-level.

Fine Spells

- This skill is also great for Sivir if you like exhaust better than Flash. The skills purpose is to escape and engage battles better.

- If you know how to use this spell well feal free to take it but there is 2 problems with this spell:
1) Sivir doesn't have AP at all.
2) By this build Sivir is an AD champion and less DPS.

- You can use this spell on Sivir but I think the effect of Exhaust is much more effective.

- I personally don't like this spell but Sivir is a squishy champion and this spell can really help with that.

- This spell can help you if there team have heavy cc.

Do not take spells

- you don't need this spell as long as you know how to use your "E".

- This spell is better left for the support in your team.

- This spell is better left for the defender of your team in dominion game mod.

- This spell is better left for the jungler in your team.

- Never ever take this spell for the reason of it is better to not die in the first place then to die quickly.

I rebuild this build and I am still working on it.