Overview of Caitlyn
Caitlyn is a ranged AD Carry Champion. AD Carries tend to have very high autoattack damage as well as stunning skill damage, especially mid-late game. Caitlyn, like all champions, has her pros and cons.
Range of 650 (compared to 550 for other ADCs) allowing Caitlyn to attack enemies when she is out of their attack range
E skill good for escaping and slowing enemy down, potentially helping yourself or your team
W skill good for placing at bushes
R skill good for killing enemies who are running away
Q skill good for harassing
At the start of the game, Caitlyn's rather low attack speed means that you have to be able to time the last hits properly
Caitlyn's mid game (around the time you build your Phantom Dancer) may be slightly weak as her health starts to get low and she cannot kill armored tanks. This is mitigated once you get Last Whisper and your Tank Item.
If you lose out at the start of the game, there is a high chance the rest of the game will be bad for Caitlyn.
Caitlyn is a good ADC but it is better to try easier ADCs like Miss Fortune or Ashe first.
Flash is essential on any AD Carry. It allows you to escape over walls easily and quickly if being chased, and can help you in chasing and securing a low-hp enemy. If your level is <12, use Ghost instead.
Ignite is good when attempting to kill enemies with low health, especially early game. This can be replaced with Cleanse or Exhaust.
Headshot is a passive skill that grants 250% damage to a minion or 150% damage to a champion every 6-8 attacks. Attacks from brush count as 2. This skill is good throughout the whole game. At the start, it provides a nice amount of damage on enemy champions if you aim them. Towards the end of the game, when you have built your core items, especially Phantom Dancer and Infinity Edge, this ability, coupled with the 250% critical damage will make for an insane hit, usually >1000 damage by mid game. Try to aim Champions instead of Minions with Headshot.
Piltover Peacemaker is the skill assigned to the Q key. It is Caitlyn's main attack and harass skill. It has a very long range and works well especially at the start of the game. However, there is a cast time of 1 second which means enemies can potentially run out of the line of fire. Because of this, it is advisable to try and predict your shot. As you are following an AD (Attack Damage) build for Caitlyn, once you get most of your items your autoattacks begin to do more damage than this skill. However, since this skill attacks multiple targets (goes through), you should still use it when faced with multiple opponents or to harass.
Yordle Snare Trap is the skill assigned to the W key. It is rather useful for all situations, but it is nothing like Teemo's Mushroom. Place traps around bushes, or jungling mobs. For crowd control or when being chased, you can also place traps in the open. Note that you can only place 3 at a time.
90 Caliber Net is the skill assigned to the E key. It is useful for slowing enemies down and since it pushes you back, useful for escaping. Mainly, this skill is used to escape as it can propel Caitlyn through walls, thus it is maxed last but a skill point is added into it early.
Ace in the Hole is the skill assigned to the R key. It is Caitlyn's ultimate skill. This skill should be added as soon as possible (levels 6, 11 and 16). The skill can damage enemies who are escaping. Imagine if an enemy has 1 bar of health left but he flashes away. Just unleash this skill on him and he is done for, unless an enemy intercepts the bullet. It is also used for ksing, to a certain extent.
Personally, I prefer Doran's Blade to begin with. It gives a +80 health boost and has a life steal of 5 per hit. However, a safer start would be to buy Longsword + 2 health potions or Boots of Speed + 3 helath potions.
Ever since the Blade of the Ruined King was buffed, every ADC should get Bilgewater Cutlass as their next item. After that, get boots of speed to counter Caitlyn's rather low movement speed.
Right after that, build Blade of the Ruined King ASAP. The Attack Speed allows you to put off upgrading boots for awhile, the active allows you to slow enemies down for chasing/escaping, and the 15% life steal allows you to quickly farm your health back. However, the AD provided is only 25 and thus the next purchase is the B.F sword to maximise AD. As you already have 15% life steal from the Blade, avoid bloodthirster for Caitlyn.
The next item you should build is boots, to give caitlyn a total of +60% attack speed. Then, build Infinity Edge for your core damage. Followed by that, build Phantom Dancer to amplify the effects of Infinity Edge and provide Caitlyn a total of +110% attack speed.
The next item you should buy is either Last Whisper or the Black Cleaver, depending on the enemy team's total armour.
The last item you should buy is for survivability.
The Frozen Mallet gives you 700 Health, 30 AD and a slow effect whereas Warmog Armor gives you 1000 health and 0.2% health regen per second. I prefer Frozen Mallet, but this is up to you ultimately.