Depends on which ADC champion you are playing really.
There are generally 3 starts:
Most ADCs go for an early Infinity Edge for the power spike.
Those that rely on AD scaling abilities such as Ezreal and Corki opt for a Trinity Force.
Hyper-carries (high DPS, % health damage procs) rush a BoTRK to get more of their procs off.
(Buy boots as you build your first item - it helps tremendously with kiting)
After that, it gets a bit more complicated.
Assuming we went with start one,
in most situations we will follow up with a Statik Shiv or Phantom Dancer. Afterwards, we would get a BoTRK if the game is evenly tied, or a Blood Thirster if we are ahead.
Then we top it off with a Last Whisper for the inevitable armor late game, and a Guardian Angel for the passive.
However, it is an option to forgo the GA for one more PD or SS if your team is really ahead or have good peel. Still, it is not recommended. Against an enemy team with a Zed or high CC, buy a QSS into a Mercurial Scimitar in place of GA or SS/PD.
For Vayne, the build is pretty much the same except BoTRK is built first.
For Kog'Maw though, most rush a Trinity Force into BoTRK then into IE, leaving out SS/PD.
Corki has generally the same build as Kog'Maw.
Lucian and Vayne can swap out PD/SS for a Youmuu's Ghost Blade if they wish to.
Ezreal builds Tri force first; however, he must buy a tear very early on to stack it into Murumana, which replaces IE.
Most ADCs should get AS boots. In situations where the enemy has a ton of CC or magic damage, ADCs can switch their boots for MR boots instead.
Corki builds magic penetration boots and Ezreal builds CDR boots since they are mostly AD casters.
Oh yes, also don't forgot to upgrade your trinket, be it red, yellow, or blue, after completing 2 to 3 items. They practically pay for themselves and saves you a lot of gold.