Just go to the site and choose the "Transfer" option from the "Domain names" menu:
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If you want to transfer a .com domain, that currently costs £9.99
You might want to have a look at their Domain Transfer Checklist page before you get started. There are different rules for .uk domains and for .com domains (dotcom domains are referred to as TLDs or top-level domains). In a nutshell, for .uk domains you need to ask your existing registrar to change the IPSTAG and for TLDs you need to ask your existing registrar to unlock the domain.
Once you've transferred your domain to 123-reg, you can use their web-based admin system to update the nameservers.
Our nameservers are:
ns0.primarydns.co.uk - 62.197.34.110
ns1.primarydns.co.uk - 62.197.35.33
ns2.primarydns.co.uk - 69.162.127.106