You cannot "sign up to Star Alliance" as they do not have a frequent flyer program. They are a marketing alliance
You sign up with an airline ffp.
Airlines have alliance & non alliance partners
Where to Credit | Where to Credit Ethiopian Airlines Flights in J Booking Class
Aegean Airlines (A3) has been popular as "hold all" Star ffp, but has lost some appeal recently

General:- Which Frequent Flyer Program to Join? Help Is Here!

Oneworld:- Deciding on a Oneworld Frequent Flyer Program? Help is here]

Star Alliance: Need advice on which Star Alliance FF scheme to join? - please post here

Skyteam:- Which SkyTeam program should I sign up for?

Be mindful of expiry Miles/Points that Do and Don't Expire. Click to open the wiki.