You will have a fabulous time. Gower is my home patch, and it is stunning and really suitable for dogs, with masses of amazing paths, commons, woods, and beaches.
Here's the link to Swansea Council website here which gives info on which beaches have dog bans and when.
There are also some lovely Forestry Commission walks in Gower, and near Swansea at Penllergaer.
Also nearby and worth a visit is Pembrey Country Park - there are miles and miles of beautiful, empty dunes and countryside, with woods as well, plus a long sandy beach. Dogs are banned on part of it, but that still leaves an enormous expanse you can walk on. The park itself is fab for picnics (it has BBQ areas) and is so big it rarely gets busy, as there are lots of lovely little corners you can explore.
Gower has some of the finest beaches in the UK. Safety note - some of the southern ones such as Pobbles, Three Cliffs etc are prone to weaver fish in the summer so don't walk on those barefoot. Most years, someone gets hurt. Also, some of the inland commons near Pennard, Lunnon, Ilston etc. have adders - not a major problem, but just be aware of them sunbathing.
Rhossili is stunning, but I love Whitford Burrows (NT) for dogs - not suitable for swimming due to currents. This means it is often v quiet in summer. Oxwich is amazing - often popular, but so big, you just need to walk a bit past the busy bits and then it's all yours.
Three Cliffs and Tor Bay are stunning.
Bishopston Valley a nice walk. Clyne Valley links Gowerton, Dunvant and Blackpill - lots of lovely walks in the wood, and you can cycle down to the prom.
Whatever you do, don't forget to have a Joe's icecream. Swansea expats practically salivate on the spot if you mention this.
It's a Swansea institution - the parlours are in Mumbles and at the bottom of Brynhyfryd Road,near the Guildhall.
You can pre-register your dog with DogLost Website - brilliant idea, as you can upload all their details, pic etc.
YY to plenty of throws and towels, and a couple of spare leads/collars in case they roll in something dead and theirs need washing. Small bottle of dog shampoo also handy.