This guide is focusing on the strategy of early lane domination, through easy spamming of spells. For an all around guide to sivir I recommend you "Sivir - All you ever need to know!" from Meridianprime, to get some knowledge about sivir and how to play her.
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Runes

Every AD Carry runes should work, flat attack damage fitting the most because of her good scaling on Boomerang Blade (0.8 ad on lvl 2 ) + strong auto attacks.

Examples for other usefull runes:
Armor penetration (= pro: higher Dmg output, con: does only passive enhance q-scaling)
Attack Speed (= pro: more Dmg over time,con: reduces early game dmg in comparison to other runes)
Lifesteal (= pro: more sustain, con: reduces early game dmg in comparison to other runes)

Masteries

Fairly standard 21/9/0 for maximum damage, while having decent defence.

Basics

1. We want to max Boomerang Blade first, because its our main source of single target damage.

2. We want to max Ricochet second, beause this is the tool that allows us to zone our enemy from the minions and shove the lane more easy.

3. We want to get Spell Shield third, because it allows us to negate most of the enemies harass. Since we are going to buy Tear of the Goddess we do not rely on the mana the spellshield would provide us with.

The only question here might be, why do we skill Ricochet at lvl 1 ?
Answer: To shove in the lane as fast as possible and therefore getting an advantage fast and early.

Tipps

Boomerang Blade

- for maximum damage hit the enemy first (do not hit minions before you hit the enemy champion)

Ricochet

- while ricochet is active you have increased attackspeed
- activate ricochet after you attacked the enemy to deal more damage( Riochets resets the attack animation )

Spell Shield

- do not waste your shield on the first spell aimed at you, comprehend which spell is lethal and therefore needs to be blocked and which one is not.

On the Hunt

- do not waste for return to lane situations ! This ability can make or break your junglers or the enemy junglers gank.

General

- while your tear is not upgraded spam abilities even out of combat to acquire stacks faster
- dominate your lane by pushing and harassing under their turret for maximum annoyance.

In the following sections I will go in detail over the item choices, especially the unusual ones like manamune or sword of the divine.
First off to note however is, that the items we choose have to work around the champion and not the champion around the items. Therefore the Standard ADC build even though it provides you maximum damage is in my opinion not the way to go for sivir(Standard ADC Build).
Why ? , because the usual ADC build does not focus on enhancing sivirs natural strengths, it is just designed to make you an hard hitting target with one of the shortest auto attack ranges ( Sivir = 500 ; Caitlyn = 650/Source).

Pros / Cons of Sivir

Sivirs Strengths

1. Fast pushing
2. Strong early game
3. High damage output early
4. Easy harassment of enemies in the lane phase
5. One free get out of jail card with spellshield
6. Excellent farming capabilities
7. Massive AOE Damage

Sivirs Weaknesses

1. Slow mobility
2. Short auto attack range
3. No reliant single target burst
4. No build in flash mechanic

Starting Items

There are mostly 3 choices which can be considered good:

Doran's Blade + Health Potion - health, sustain, damage, actually all you are looking for; except for mana, which you will need to play in easy mode!
Boots + 4* Health Potion - increased health regen through pots + higher mobility, which gives you an easier time farming
Faerie Charm + 5* Health Potion, 3* Mana Potion - the easiest way to start out the game, because this allows you to spam the heck out of your spells.
Since we want to play easy mode we naturally decide to go for this starting items choice.

First back to base

Tear of the Goddess + (2*) Doran's Blade is what we are aiming for. Boots wont hurt either.

Second back to base

Now we want to maximize our early game strengths. Therefore we acquire Vampiric Scepter, Berserker's Greaves and Long Sword or complete our Manamune. The Scepter should increase our lane sustain, berserker's Greaves enhances our damage over time and long sword or manamune gives us the attack damage needed.

The inventory should now look something like this:



or this:

After the Early Game in which you probably should have dominated now the time has come to acquire your Core.

Manamune

This item is considered core, because it is giving you all the early game pressure you ever dreamed of having on an ADC, while not relying on the use of spellshield to refreh your mana pool( Using Spell Shield to sustain mana is even dangerous, because while its on cooldown the enemies have an 20 second window of opportunity to move in and kill you).
Even though late game there are items that could do better for you, like Infinity Edge, its the Tear of the Goddess which should have had most impact in your game, by giving you the edge in every lane matchup through superior harrass power. Late Game Muramana is also not bad to have, even though other items could outclass it.

Bloodthirster

The most Damage you can get in a single Item and topping that off with nice, solid 12 % life Steal. You want flat Damage as Sivir to scale your abilities with. So this Item is a must have as Sivir, always!

Sword of the Divine

Probably the most controversial item choice. It provides you passivly with non reliant attackspeed. Well non reliant ? Yes, should you decide to use its active ability the attackspeed provided by sword of the divine is gone. So why should we get this item over Phantom Dancer or Statikk Shiv wich even provide us with increased movement speed ?
The Answer is not that easy;
Let's look at the pro's and con's of sword of the divine

Pros:

1. Enhance's Attackspeed
2. The item is with 2200 gold cost the cheapest attack Speed - item available
3. Increases your potential to burst
4. Gives you 3 guaranteed critical strikes
5. Synergieses godly with Ricochet

Cons:

1. Its passive does not give you any boni besides attack speed
2. While on cooldown your attackspeed buff is gone

So these things considered you are after acquiring Sword of the Divine not the usual ADC anymore. Now you have burst Damage, for the price of lower movementspeed and decreased damage over time. Is this a trade off you would want to take as any ADC ? No !
But Sivir is not any ADC, the lower movement speed should not greatly impact you cause you always have On The Hunt for kiting in teamfights. The Damage over Time decrease also is not a big problem for Sivir because after activating ricochet in combination with Sword of the Divine you bursted one target and applied huge damage on the enemy team within a few seconds, so your job in the teamfight is already done. Since you bought manamune in the first place you are 1 item short that would have given you Critical Strike chances. Sword of the Divine makes use of this by providing you guaranteed Critical Hits without wasting any Crit- Chances you might have had from items bought earlier.

Berserker's Greaves

Easily the best boots available for an ADC. Only get Ninja Tabi if facing a full Ad team or Mercury's Treads when facing at least 3 hard hitting enemy casters. Do not get it if you are afraid of stuns and snares, prefer Quicksilver Sash or Zephyr over Mercury's Treads.

As stated before Sivir is not the usual type ADC. She is one of/ if not the best pusher early on. For this reason you should play sivir not so much as usual ADC, but more to her strengths. This implies pushing 24/7 if given the opportunity. The recommended build above emphazises on the aspect of pushing and harassing in your lane.

So how exactly do we push and harass as Sivir early on ?

Starting off with Faerie Charm and mass potions, we rush on lane and use ricochet before the minion waves clash in the middle of the lane. Now we continue without missing many lasthits to push our wave into the enemy tower. At around 3 minutes we ask our support to set up vision for us (If he does not, use your trinket). Vision is a crucial aspect of manamune Sivir since we are constantly in enemy territory in front of their tower and therefore very vulnerable to ganks.
Granted you achieved to maintain vision, you should continue to siege the enemy tower and trade damage with your lane opponents( Remember: You started with 5* Health Potion so you can regenerate health faster and more consistent as your opponent. As soon as you drop hitpoints instantly use your potions. Do not stay in a pushed lane with only half of your hitpoints remaining!).
Applying the described strategy will leave your oppostion usually with half their hitpoints, while you have taken hardly any damage ( or regenerated it quickly). In this stage you should now drop all aggression and try to freeze the lane in the middle or in front of your tower and ask your jungler to come bot for some easy kills. ( Do however trade damage if you have the possibility to.)

Playing as Sivir in later stages of the game:

Assuming you have acquired your core items, team up with the other champions and start siegeing a tower. Harrass with Boomerang Blade and Ricochet as much as possible; mana - problems do not occur on manamune Sivir.
If an enemy champion drops low on health during the siege, ping your team to attack, activate On The Hunt, Ricochet and Sword of the Divine and delete your target. After deleting the first enemy reevaluate the situation and try to determine if your Ricochet bounces did enough damage to continue a fight under the hostile turret. If you came to the conclusion diving for more kills is not an option, ping your team back and continue the siege.
( Remember: While Sword of the Divine is on cooldown your Damage output drops significantly.)

In this chapter I will list the support that will come in handy to have on lane with you.
Generally speaking you want a support which is good at the same thing as you are good in. For Sivir this means you want a support, which can help you push, keep you safe, provides you vision and most important can help poking your opponents. Therefore all in supports, like Leona are not as good at supporting Sivir as at supporting other carries.

Godly Supports

Good Supports

Okay Supports

Foes

Manamune Sivir does not have any ADC - opponent outclassing her. But she will have a hard time against lanes, that can set up ganks easily and farm exceptionally well under their turret or poke her. Manamune Sivir also is very vulnerable to mobile junglers spamming ganks on her lane.

Natural Enemies

Believe me, these boots on a skilled jungler can make your life Bottom Lane pure hell.

Assumptions:

Build 1 follows the Manamune + Sword build.

Build 1 buys 2 Doran's Blade he will need to sell in order to complete his build.
He will go 2 times back before 15 minutes to purchase his early items and pick up 2* Health Potion every time he backs.


Build 2 follows a " normal ADC - build".

Build 2 starts out with 1 Doran's Blade he will need to sell in order to complete his build.
He will go 2 times back before 15 minutes to purchase his early items ( Vampiric Scepter + Boots or B. F. Sword) and picks up 2* Health Potion every time he backs.

Build: Potions before 15 mins (2 potions per back to base/2 times back) Gold lost by buying and selling Doran's Blades "More" Gold cost total: All in all price:

SPACESPACE

Manamune + Sword Starting out: 5* Health Potion 3* Mana Potion 4* Health Potion (by back to base) = 12*35= 420 Gold 2* Doran's Blade = 528 Gold 420+528 = 948 Gold 15423 Gold

SPACESPACESPACESPACE

Standard ADC Starting out: 1* Health Potion 4* Health Potion (by back to base) = 5*35= 175 Gold 1* Doran's Blade = 264 Gold 175+264 = 439 Gold 16214 Gold

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Muramana + Sword Build is 800 Gold cheaper. It's safe to assume however that 800 Gold for finishing your build will not affect your game in any major way.

Items at different time marks

The following Items one could have farmed under the following conditions:

You missed every 5th Last Hit (80%).
(Gold over time, minions additional value over time and starting gold included)

10 minutes:
total Gold: 3260
flawless farming: 3704





20 Minutes:
total Gold: 7256
flawless farming: 8413





30 Minutes:
total Gold: 11992
flawless farming: 14060





37 Minutes:
total Gold: 15530
flawless farming: 18302

Conclusion

The trade off between Manamune Build and a "Normal" Build is basically around 30 damage early on ( minute 10 ) for unending pressure by spamming your abilities. And if this pressure does not lead to an advantage for you, you are basically almost even with the normal ADC - Build in effective combat stats. It is on you to decide if you like to lane with unending mana and unending opportunity or try to start the usual ADC arms race.

Skill order:

R > Q > W > E

Start out with:

Faerie Charm + 5* Health Potion, 3* Mana Potion

Get ASAP:

Tear of the Goddess + (2*) Doran's Blade

Aim for:

Manamune + Bloodthirster + Sword of the Divine + Berserker's Greaves


DOMINATE !