How to keep my stainless steel skillet clean
Be careful with cold water in a hot pan - since the pan is metal, you shouldn't have to worry about cracking, but you will get a nice burst of steam in your face!

Your best bet is to deglaze the pan immediately while it's still hot, or heat it back up and deglaze it.  There are some tips in this thread.  As a bonus you're going to get a delicious pan sauce as part of the deal.

You're never going to get it store-bought perfect, but I've had a lot of luck cleaning stains that seemed permanent by using Bar Keepers Friend, which in my opinion is one of the best commercial cleaners available.