Do dungeons or other stuff with random people, when you find someone you like or had fun with, send them a friend request. Two clicks! A lot of the time it's that simple. People don't even have to be on your server for this anymore.

Otherwise the #1 way into an FFXIV player's heart is to compliment their glamour (or their house). Nearly every non-FC/RL friend on my list got there this way.

Duty Finder conditions people into thinking that nobody ever wants to talk, but the opposite is really true -- it's just that pretty much everyone's been conditioned in the same way. It can be scary to initiate, but somebody has to take the first step.