Girona. Never been there myself, but it's on the list. More of an emphasis on the cycling side than the other stuff, but I think it'd tick all the boxes.
Where I'm actually going for the second of those weeks is Cesenatico for a couple of days and then the Dolomites for the rest of the week.
13th of May to the 27th - where would you go in order to enjoy (in no particular order):