If I wanted to try and do a trip like this in England, is the best way to do the transportation to rent a car in London? Or could I take a train for the entire journey? Traveling with a family.

If I wanted to try and do a trip like this in England, is the best way to do the transportation to rent a car in London? Or could I take a train for the entire journey? Traveling with a family.

You could train that, but as soon as you cross over into South Wales the train links deteriorate and you might find yourself needing to make a lot of changes to get to your final destination.
Think you might have to change in Swansea. Long time since me caught the London to swansea train. Trains in UK are very expensive too. Might be cheaper to rent a car. Plus a car would save the hassle of changing trains, waiting, and the possibility of cancellations. Cancellations are rather frequent. Good luck and enjoy your trip.
If you can make a short detour and visit Gower. 20 odd miles outside Swansea and, if me remembers correctly, the first area of outstanding natural beauty in the UK.