Choosing a custom slug
One slug, one link - across golo.to and your own domains.
Set a custom slug the same way regardless of which domain you're using - see How to customize a slug.
The one domain-specific thing to know: your golo.to links also work on your own domains, so within your account a slug points to one link. golo.to/deal and links.you.com/deal are the same link - you can't give them different destinations.
If a slug you want is already taken on golo.to by someone else but free on your default domain, GoloLink tells you ("Taken on golo.to") and creates the link on your domain only.
Related articles
Adding a custom domain - Type it into Settings - then complete DNS setup and verification.
Connecting a custom domain - Add your domain and verify it with DNS records.
Creating a branded link - Pick your verified domain when creating the link.
DNS setup - One TXT record to prove ownership, one A or CNAME record to route traffic.