Jack, you've got to go to Berlin, London, Dublin, Paris, Barcelona, Madrid. These are all my favorites. Berlin for a month costs me the same as London for a week, yet they are equally great world-class cities with a lot to see and do.
Look for a good economy fare on Kayak into Dublin which is always cheapest, then fly cheaply to any of the others using Kayak, or take the EuroRail and see the countryside across the continent. Walk the cities or hop the buses which go everywhere all the time.
You can stay cheap in people's homes in your own bedroom using AirBNB. I stayed in Berlin for a month last summer at a student's place on a shady cobblestone street for $26 a night. He was a great host (has two other apts he rents via AirBNB). When I fixed his laptop he cooked German food for me the rest of my stay. He's going to give me a monthly rate directly next year.