Random invites are kind of annoying, if I am out grinding around and happen too see someone at a fate that we are both participating in, after the fate is over I send a tell asking if they would like to join an FC, explain a bit about FC benefits and our group and let them know it's ok if they don't want join then send an invite and head out. But I do remember being a Greenleaf and getting bombarded by FC invites, it gets old fast. Dont be afraid to join one as you can leave at anytime if you find another group fits your playstyle or needs.
Have fun, and welcome to Eorzea!