Unfortunately the Mochyn Du is no more. It’s reopening as the Bragdy a Chegin, although it’s unclear how Welsh speaking it will be yet.
In Treganna (Canton), Grangetown and Pontcanna there are a few places that have Welsh speaking staff. Canna Deli is a lovely café/deli and the clothes shop next to it is also Welsh speaking, and Caban (just over the road) is a welsh language bookshop (all in Pontcanna). A few of the cafes and pubs on Cathedral Road have the orange speech bubble up (which means they speak Welsh there). The Grange Pub has some quizzes etc in Welsh, the Cornwall pub also does a Welsh night on a Wednesday (both in Grangetown). The Lansdowne has some Welsh speaking staff, Chapter Arts too, Capel Salem is a Welsh speaking chapel (I’m going to claim them for Treganna before the Pontcanna crew try and have their say 
). Alys Williams and the lead singer from Candalas are in concert in one of the other Churches soon. Iechyd Da is a Welsh speaking gym in Treganna too.
The Hen Lyfrgell in town is pretty good for Welsh service (though not guaranteed). Bodlon in Eglwys Newydd (Whitchurch) is a lovely café and they have a tit bits shop there and under the Hen Lyfrgell in town. Cant y Mil on Whitchurch Road in Mynydd Bychan (the Heath) is a Welsh speaking shop.
The Cadwalader’s Ice Cream shop and cafe in the Bay have the speech bubble too.
So there are a few places to start (this isn’t an exhaustive list). Menter Caerdydd e-mail out regularly with events, that’ll give you a clue the way that Siaron has suggested.
It depends what you’re looking for too. There are rugby clubs, sports teams, pwyllgor (raising money for the Eisteddfod), and there’s still a nightclub. Around Treganna/Pontcanna, you regularly hear Welsh in cafes and pubs just from groups of mates. A lot of the 40,000 are ‘hidden’ because they have jobs that aren’t in the service industry so they just use the city the way anyone else would. So a lot of it isn’t easy to find because you’d have to go to hospital, school, Admiral Insurance or wherever to find them.