I would say May is almost summer. You can have some cool days, but most will be Spring/early summer ones.
I would bring summer clothes and a couple of pullovers and a raincoat just in case.
Most girls use purse, specially when going out, and specially if trousers don't have pockets. In this case, a small wallet wouldn't help much unless you'd be wearing a jacket and put in in the interior pocket. For security measures, there is plenty on info on the board about ways to carry money (unnoticed) on you.
Many girls use to carry a small wallet into their pocket, but this is not a very good place to carry it for a tourist, since if you are identified like a possible tourist, your purse is in special danger, specially in touristic areas.