I've taught すごい to friends as their first or second word too, it's very useful. I wouldn't say that it means "cool" though, more like "wow!". You can't use it to say "a cool guy". On its own as exclamation it means "cool" (like when you're looking at some great scenery). It's a little informal when used on it's own like that. You can definitely use it in a less informal way, but the meaning there is "very", not "wow!".
昨日はすごく寒かったです。 - It was very cold yesterday.
I think to make it even more formal, you can use 大変
昨日は大変寒かったです。 - It was very cold yesterday
I don't know if the strength of the coldness changes between them. 大変 might sound a little stronger.
The other meanings are "dreadful" and "horrible", or really anything adjective that you want to make stronger: "tasty" -> "delicious" (すごくおいしい), "beautiful" -> "gorgeous" (すごくきれい), "loud" -> "ear-splittingly loud" (すごくうるさい).
For example すごく怖かった - Very frightening.
The slangier examples are すげえ, which is kind of masculine, and すんごい, which adds (even more) emphasis. You can extend the え in すげえ (すげえぇぇぇ) or extend the ん in すんごい for added effect.
Edit I realise that I didn't actually answer the question. Like Mark, I have trouble differentiating between slang and informal and even though I said "slangy" in my original answer, I think that using it to mean "wow!" is just informal, not slangy.