Sugoi (すごい) means “amazing”, “great” or “incredible” — and more broadly, anything extreme, good or bad.
You learn すごい in week one. Then a character says すげー, another says すっげー, someone else shouts うめー at a bowl of ramen, and none of it is in your textbook.
One sound change, a whole vocabulary
Casual speech squashes the ending of い-adjectives, and it does it by rule rather than word by word. あい and おい collapse into a long え — that's すごい → すげー, たかい → たけー, やばい → やべー, うまい → うめー. うい goes the other way, into a long い: さむい → さみー, わるい → わりー.
So once you hear the pattern you can decode words you have never met in that shape. It is rough, male-coded and very casual — great for understanding anime, risky for speaking. Nobody minds if you never produce it; everybody notices if you use it at work.
The bit even native speakers blur
すごい is an い-adjective, so describing a noun it stays as it is — すごい ひと. But in front of another adjective or a verb it should become すごく, the same く as in やすくて (Chapter 8 of the course): すごく おいしい, すごく つかれました. In casual speech you will constantly hear すごい おいしい instead. It's the Japanese equivalent of “real good” — everyone says it, and it's still worth knowing which one is the tidy version.
この ラーメン、 すげー うまい！
This ramen is SO good!
…すごく おいしい、 ですね。
…very tasty, isn't it.
What should you say?
すごいです or すごく + adjective will carry you anywhere. Keep すげー in the listening half of your brain, where it belongs — and note that すごい, like やばい, is not always praise: it marks anything strong, good or bad. すごい あめ is a downpour, and すごい ひと can be said of a crowd.