Well, I have to confess I don't really go into Carmarthen a lot, and when I do I rarely go in a café so I'm not the best person to recommend anything. When I do meet people there, I usually suggest Yr Atom in King St, which is the Welsh Language Centre for Carmarthen http://www.yratom.wales/yr-atom-cafe/ so I like to try and support it.
However, it doesn't have a lot of space, and tends to get quite busy. I haven't been there on a Saturday though, so I'm not sure what it's like then.
There is a Café Nero right in the centre of Carmarthen, that has more room and if people wanted to use that as a meeting place then little groups could have a wander about, visit the castle etc.
I would love to say I would be there early and meet everyone, but I have my elderly parents arriving next Wednesday to stay for 6 weeks and what I can do during the day on the Saturday is going to depend very much on them.