I am thinking "C-infinite" or "infinitely differentiable". The second one is absolutely correct, but too long for colloquial use. I'd prefer the first one if possible. Is it correct or common?
So long as it is clear what $C^\infty$ means in the context where you're speaking, it's perfectly valid to say "C infinity." Otherwise, you might want to establish what you mean by that, but then it should be fair game.
Granted, in my opinion, to say $f \in C^\infty$ as "$f$ is infinitely differentiable" just rolls a little nicer off the tongue. But this is a personal quirk - so long as it's well-established what you mean, it's just a matter of personal preference.