At the beginning it is important not to spam your abilities and save the mana and the readiness of your abilities for the opportunity to get a kill or two. Just last hit with basic attacks and stay well into your minion line. Become aggressive, then your health pots will start being used rapidly. Communicate with the support and work with his or her abilities. Blitz crank and thresh for example can bring the enemy to you and when that happens unload your attacks as fast as you can to get the kill; ignite only to ensure kill. Most importantly play defensively and take advantage of opportunities during the beginning

In team fights, Sivir needs to be there as soon as she has the first two items built (blood thirster and infinity edge). beginning boots and the third item only helps. During team fights it is important to lay as much damage to the adc first then focus other high priority targets. If the battle is in your favor and enemies start to run pop your ult (its called the hunt for a reason) and take down the fleeing champions. However if the battle is going sour, pop your ult to escape, preferabbly with as many teammates as possible (ult works on teammates too).

When coming into a team fight, I initiate with Q then follow it by W, all the while waiting to block enemy ults with E. By the time the battles conclusion the cool down for Sivir's Q should be completed and use the Q ability on a fleeing, low health champ to secure the kill.

Her Spell Shield (E) is simply amazing. Facing champs that have slow ultimates such as Karthus, Caitlyn,Lux,Fizz, etc. becomes a breeze with Sivir's Spell shield as it blocks, completely, the next enemy champion spell that is cast, and if you save her E for those ultimates it can save you form dieing or from losing vasts amount of health. BUT your timing has to be spot on. Once mastered, however, it will be much harder to kill you.

Upon completion of the 2nd item, killing champions should come much easier since your damage output is there. Once the 3rd item is complete, I find that Sivir begins to snowball, becoming only stronger due to her ability to kill enemy champions, including tanks.