To help you to have a "care free" holiday with your dog, the main points are summarised below. There are also a selection of links to help you in planning your journey to Wales and days out.
Before the main holiday season - generally 1st October to the 30th April - there are hardly any restrictions, other than keeping your dog on a lead on promenades and being expected to clean up any "mess". In the Summer dogs are allowed on the beaches outside certain restricted areas - maps illustrating these are on notice boards at beach entrances.
The Pembrokeshire Coast National Park has made great efforts to make the Coastal Path more "Dog Friendly", Pretty much all of the Pembrokeshire coast path is now stile free, making it very accessible to dog walkers. Carmarthenshire has no stiles along its length of Coastal Path, but Ceredigion has.
There is a warning about taking your dog on any Coastal Path that you would be wise to heed:
"There is a real possibility that you will come across both wildlife and farm animals on the coast path, so you will need to keep your dog under control. The coast path can also be a dangerous place and an uncontrolled dog may lead to accidents as the path, in some parts, is very close to the cliff edge. The Coastguard advises that owners should keep their dog on a lead; once a dog picks up a scent, that dog has no thought for its safety!"
Pembrokeshire, in particular, has miles of footpaths and bridleways and Carmarthenshire is not far behind, nor is Ceredigion. In the main you are free to roam, but many rights of way cross farmland and it is necessary to keep your dog on a lead if there are sheep or cattle grazing. Dogs are not allowed into the nature reserves or on any of the Pembrokeshire Islands.
Ceredigion's advise is: "Generally you can take the dog along, unless there is a restriction stopping you from doing so." However, certain paths can be restricted at times: "dogs must be kept on a short fixed lead at all times whilst on permissive access routes or areas and that access for dogs may be stopped altogether at certain times of the year".
There are many upland areas worth exploring with your dog - the Preseli Hills are probably the best known, but there is also the Teifi Pools, an area of moorland remote from civilisation.
Riverside walks that come to mind is the footpath along the Teifi from Cilgerran to Llechryd (you can let your dog off the lead here), the Gwaun Valley and along the Cleddau estuary.
The Forestry Commission has many acres of forest in Ceredigion & Carmarthenshire. There are literally hundreds of accessible footpaths. For example it is possible to combine seeing the Red Kites being fed with a decent lakeside walk at forestry feeding site of Nant y Arian.