Sivir is probably the best AD carry atm, and I suggest anyone to learn how to play her. Well timed ganks and skills shots will win almost any team fight. This is mostly to give a general idea of how I play Sivir and why I play her this way. I don't expect this guide to be the most informative but more or less just getting to the points.

Pros:

Amazing Early and Late Game Champ.
Best Pusher.
Well timed E's can make for a Epic Team Fight.
Great Mobility with Ulti, Aura makes for great Team Fights

Cons:
Squishy
Need Enemy Blue to sustain Mana

My name is Kraffty, I am not ranked by any means but am working my way up for a team. I am introducing one of my Favorite Range AD Carries and probably the best atm. I see a future nerf for Sivir because she is to what most might say OP, Yet I disagree she isn't OP just if well played an amazing carry. Well timed E's Ulti's and spamming the right Attack sequence is win. Keep in mind you should never stand still always stay on the move, this way you can avoid skill shots from champs such as Cass, Lux, Ez, and Ahri even in team fights this can be your death. Stick to the back and wait for the appropriate moment to iniate into combat, if the fight looks like its going to be a win go for it if someone dies immediately on your team your, best bet is to get out imo. Now I think one of the best choices for a bottom Lane support for Sivir is Leona, makes for an amazing kill lane. If Leona knows how to play aggressive and time her skills accordingly.

Your Q this is important, look for opportune times to use your Q, if you have no minions waves in the way and a clear shot on the opponent use it, your damage will put the extremely low, and this is why I suggest AD for Marks, and Quints.

Lastly I want to apologize it may not be the fanciest of guides, but as I said its to the point. I can't tell you how to play your champ, everyone has a learning curve that they must get through. I can only make some helpful suggestions.

Here is an image of a Penta Kill I got with Sivir using this build. Keep in mind I am not full build but just about there.

Runes, this is simple I have noticed a lot of people using Armor Pen, and such. This is ok but to me pretty worthless early to late game, the extra AD with a Dorans Blade puts you at around 80 attack damage to start out. Which means massive damage for early game feeds, with a good bottom lane support. The armor and MR helps mitigate some early game damage as well, which makes things more effected.

This is important as well, start with Doran's Blade, most start Boots pots, but getting 2-3 doran's blade early with boots makes for a bit more damage extra HP, and mobility early game. Why do I say Berserkers Greaves, well your an AD carry Attack Speed with Life Steal makes for amazing fun :P... With this you have the reduced time on slows, and the early game MR, and Well you can make up for the attack speed with Zeal. Yet if you aren't seeing yourself getting into this kind of problem make your way towards Greaves. Bloodthirster, this item is important for Sivir, make sure to farm and cap it to 10%, when you find yourself dueling it helps a lot with a Zeal and your ulti almost impossible to kill.

B.F. Sword then Negatron Cloak, I think is pretty self explanatory, Extra Damage and Magic Resist which both help you build into a core item later on. With an I.E. and Banshee's Veil, massive damage and using your E with the Passive from BV helps you mitigate some more spell damage. Phantom Dancer is for the extra Crit, AS, and Mobility. Now why a last whisper, well it comes down to you didn't runes for Armor Pen so this helps its a cheap item that gives some great states imo.

I chose Flash and Ignite for this reason, I normally lane with Leona or a very tankie support who has Exhaust Heal, with the slow and heal makes tower diving worth it.

So, this is a fun tactic to use, if you start purple and are bottom lane which you should be with this build, you want to do this. Go get there Golems, or if you start blue make sure to get golems. It is important that you do, more so because it gives you some early experience and puts you ahead of the competition. Then get in lane and make sure that you are last hitting minions not to push. This allows you to get your opponents out of position, for early kills.

When you or your oppenents turret go down this will basically be the end of your laning phase. At this point I suggest that you start farming the jungle but don't put yourself out of position either. Farm wraiths, get Red buff, steal the enemy blue, and clear minions waves. It will ultimately put you ahead or atleast stay up to par with CS, and helpful to buy items late game as well.