There are many places where you can easily launch your kayak, though not all have life guards, so please take every precaution, in particular do not kayak alone on the sea - conditions change frequently. Good places to take your kayak by car with easy access to the beach are:Aberporth, Llangrannog, Cwmtydu, Poppit Sands [bit of a walk at low tide] and Aberaeron.
You can hire paddleboards, kayaks, sailing dinghies, windsurf boards from Cardigan Bay Watersports Centre in New Quay, west Wales. www.cardiganbaywatersports.org.uk
One of my favorite explorations of the Cardigan Bay coast is to take your kayak from Llangrannog and head south to the beach at Cwm Tydu with it's lovely little cafe. Along the way you will see coves which are difficult to access from the Ceredigion Coast Path. These offer the chance to enjoy a swim and the opportunity to spot the many hunting birds above the cliffs.
Why not take a break on Penbryn beach where a short walk up to the National Trust cafe makes a welcome break. It is possible to drop your kayak close to the beach here, but it is a very narrow road and the turning place where you can unload has to kept clear for emergencies. Please download a copy of the Ceredigion Marine Code [PDF] which details how to protect Cardigan Bay dolphins, seals and porpoises.