I know a bit more about Salsa in Cologne than Tango. There are several places that regularly have Salsa nights, listed
here or
here, but the ones I know of:
Herbrand's: Thursdays is a one hour beginners course, with a normal Salsa night right after. You'll find all levels of dancers in a fairly comfortable environment.
Petit Prince: More like a basement night club, so it can get pretty crowded, but some people prefer this atmosphere. They also have a short beginner course before the normal dance.
Kultor 5: I've never actually been here, but this place is well known for Salsa in Cologne. The have dances Fridays and Saturdays, and courses during the week. Apparently there is no open bar, so people just bring along what the want to drink.
I took a Salsa course with the University. There are several every semester, and while they are cheaper for students, others can also enrol. You can attend the first course without enrolling, and perhaps find a partner. For the next courses, though, you have to be enrolled.