Dave I ride most Saturdays with the mudcrabs, rides are always featured here, usually the veloway then Willuga or clarendon return.
Check out Cancer Voices rides, again featured here, which are on a Sunday. My partner works shifts which means I can ride most Sunday mornings, with every alternate Sunday being a slightly longer ride.
Robert Rau does a Tuesday night ride from Hurtle Sq.
Lots of rides available.
Also put up a ride yourself, others will more than likely join in.
We have a Tue night group ride which meets at the SE corner of Hurtle Square at 7pm, has fast & slow groups and does various 2-hour hills rides, meeting back in the city around 9. Apart from the social aspect, you'll see various good hills routes near town, too.
You could also try the Beach Streak Thu night ride, which used to meet in Henley Square at 6:15pm and do 30km along the beach in a couple of groups. But it's been ages since I've done it, so it may have changed or gone.
the easier to understand BikeSA rides program is here Many of the BikeSA rides are slowish, with the exception of Cobwebs (Sat am), Tues night and Thurs night (as mentioned above)
I would also suggest you read through the Cobwebs Ride Guide at the same link to get an idea of the calls/signal/etiquette if you are new to group riding.
these rides are all for riders with road bikes, so I am assuming that is your present!
The Tues night ride also requires really good lights as hills decents can be fast. I use Ayups (narrow beam option).
+1 on getting insurance cover.
Next Tue night is on, although numbers may be down. There are even a few turning out for Megabike on xmas morning! I might do that and then some hills to earn my xmas lunch.