You can totally go to nakedbus and things like Nationalexpress and if you get in reasonably early (I booked around now for about the same time) you can get things like 1 pound bus trips to places. I got to Bath on a pound and it was awesome. Just to offer an alternative point of view on Cardiff, it's not hugely big, doesn't have much to do other than a Castle (which is totally awesome and you must see it) and smells like pee. I wouldn't recommend more than a night or two there. The hostels are also kinda crazily expensive in comparison to like everywhere else.

Bristol is also like less than an hour from Bath, which in turn is a little while from stone henge (apparently don't do the festival thing there it's all drunken idiots?) The British Museum is amazing, and the science museum is too (they have the particle accelerator Rutherford used! I may have geeked out, like, a lot) and right next to it is the Natural History museum which has a whole blue whale. I was slightly resentful at the idea of giving Westminster Abbey my money but omg the history. The free audiotour is really excellent and as a total history nerd there was a lot of omg Elizabeth I! William Cecil! Charles Darwin's burial!

I stayed at about 3 different hostels there so if you want I could find which ones they were and whether to avoid.