Yea it would be cool if you could setup guilds like in WoW or something. If you were in a guild it could append a prefix or a suffix to your profile name. That would be cool. I think now it's just a matter of a friends list and a member web site probably.
Blizzard is supposed to make some sort of clan support in their next patch via chat rooms i think. People won't know unless you have tags in your name. For me its just hanging out and playing with guys and voice chat via ventrilo.
To officially join a clan you usually have to go to that clans website and go through some sort of an application process. As the others have said for now there is no way to show you are in a clan besides chat. I hear with the next patch that should change though. If you are looking for a clan don't miss Fever clan's recruiting thread on these forums: http://www.starcraft2forum.org/forums/showthread.php?t=17894
