If you are going to be doing "normal" tourist activities then good running/walking shoes will likely be your best bet. If you are going to be doing any kind of real hiking then you will also need to have decent hiking boots that have good foot protection and ankle support.
For clothes, May is a nice month but you will need a rain proof jacket, a good warm jumper or fleece jacket (for the tops of the mountains and the evenings), and long trousers. The key to clothes in the mountains is to wear many light layers... that way you can take one or two off when warm, and add them again when you get cold.
Hope it helps.