imo you should go with northern europe....ie england, scandinavia, germany, etc. stick to civilized countries because some of the southern ones are really crappy if you don't have a lot of money. stay far far away from eastern europe. i'm sure the countries can be nice, but there is a lot of poverty and you run a large risk of being ropped, cheated and conned. and stay away from france unless you wanna be called an american cheeseburger heh, they're so weird . almost all northern europeans are very nice towards americans and foreigners, those who aren't are a by far minority.
just be aware that you can't just walk down to a pizza store or some cheesy restaurant around the corner and get a job at min. vage and no tax reports (or however you say it), at least i've never heard of that happening a lot, first of all there aren't too many of such places as there are strict rules about hygeine, standards, taxes, etc. you definitely need to learn the language of the country you want to live in (which is why england would be smart) if you want a job. LOTS of people speak english in scandinavia and germany though, so you're in no trouble at all if you're just on vacation and want to have fun. i'm sure you'll get loads of chicks at clubs etc if you're some handsome american on vacation heh denmark has a drinking age of 15, go nuts
i'd also advice you to stay in the capital or at least a very large city. areas far away from cities, can be very very dull, this goes for most of europe. boring old people everywhere, no nightlife and *nothing* is open after 6pm. in scandinavia, supermarkets close around 8-9pm, special stores (electronics, clothes, etc.) close at 6pm, and all that's left after that are 7-11's and lots of small drug stores with candy, some canned food, toast, and maybe a bit of milk/butter etc.
i'd say one of the things i dislike the most about scandinavia is the fact that, as said, stores close damn early. second, obtaining.....stuff.....generally is harder. you can't use coupons and buy stuff incredibly cheap on the web, the shipping fees are a bit harsh. but most items can be ordered from an electronics store, of course.
good luck more info about denmark
here