It's a nice idea and as an atheist I am also 100% for space exploration and colonization, because I think that's simply the drive behind all life forms.
Your comments/rationale about humans "bringing with us our plants and pets" and somehow filling the universe with our planet-bound companion species, though, I don't share. As an ideal yeah, it'd be cool. As a plausible outcome, I don't think so.
There's this principle that says that the most variety is always found at the point of origin. Even if we actively and consciously carry other species with us, it will necessarily be an absurdly tiny fraction of the total set, completely non-representative of Earth, and most likely based one purely utilitarian reasons.
As a matter of fact, I even think we will not carry ourselves (as in, Homo sapiens) away. It will be our non-biological descendants who will physically leave the gravity well. Our physical bodies are bound to it (unless we somehow manage to skip/turboaccelerate millions of years of evolution through conscious gene manipulation).