To sign up is to put your name down to join something. A course, a gym, a mailing list.
It takes for, and the noun signup is now standard in anything online. The word implies a commitment rather than an enquiry, which is why the sign-up form is the point every website is trying to get you to. There is a sporting sense too, since a club signs up a player. Register and enrol are the formal equivalents, and enlist is the military one.
1phrasesign upTo enrol or agree to take part in something.
“If you want to join the choir, you need to sign up at the front desk by Friday.”
Also accepted in games: register, enroll
Practised in Phrasal Verbs