Getting gear in World of Warcraft, although is incredibly easier than it used to be, is still not an easy task for a lot of people.
Most are able to get all of their 5 man Heroic gear and collect enough Triumph Badges to get around 5100 gearscore without ever going to a raid, but if you want to be the best there's no way around raiding.
Once you've obtained your 5100 gearscore, or possibly just 5000 gearscore, it is time to start looking for pick up groups.
What you are mainly looking for is pick up raids where item level 245+ items will drop.
This starts at Trial of the Crusade 10 Man Heroic, Toc 25 Regular and Heroic, Onyxia, Icecrown Citadel all versions, and the new Ruby Sanctum; in rare occasions a ToC10 Man regular with item level 232 items can be pick up grouped if in need of a trinket or some random slot upgrade.
Start with Heroic 10M TOC, Regular 25M, and Regular 10M ICC.
Once your gear is good enough you'll be able to watch for Heroic 25M TOC, Heroic 10M ICC, and 25M ICC.
From that point on you will be over 5500 GS and can probably get into 25man Heroic ICC.
You will also find yourself getting into the 10 or 25 man versions of Ruby Sanctum occasionally.
The key thing to remember is different servers and pick up groups randomly require different gear scores and achievements to get in.
If patient enough, it is possible to get into most of the raids with a lower gear score than is typically expected but this also means the fail rate could be high for the pick up group as the candidates are not being properly screened.
When push comes to shove, there's a huge difference between slight increases in gearscore once passed 5000 and it can make a world of difference on your healing, tanking, or DPS; more than most think possible.
So begin the raid gear progression as soon as possible.