In most cases, I'm already watching the weather and know what's going on by the time the sirens go off. If we're in the path (or possible path) of a tornado we'll be hunkered down in an interior closet (no shelter, alas!) or if its off somewhere else and going a different direction we just keep watching.

In the rare occasions where I'm woken up by sirens its usually a bit more of a mad scramble to both figure out what the situation is and also get somewhere safe in case there's not any time. So far, though, that's only happened when its elsewhere and I feel foolish for panicking a little.