It is not necessarily easy to move here. Without money or a job, it would not be possible if you're not an EU citizen and not feasible even if you were from the EU. You can only be here for a maximum of 90 days as a tourist without a visa. Depending on where you're coming from, you may need a visa even for a short visit as a tourist. Also, you would need to speak the local language fairly well for most jobs.
Here in Europe, citizens of the EU have the right to live and work within the EU. It would be illegal for others to do so without a proper visa. The site for visas here in Italy, for example, is:
http://www.esteri.it/visti/index_eng.asp . The rules in much of the EU have been harmonized and will be similar. You can also check the website of the consulate that has jurisdiction over you for any country that you might be interested in.
A work permit is separate - you cannot apply for that yourself. The company has to apply and they have to be able to demonstrate that there is not a viable EU candidate for the job. As a result, jobs for foreigners including US citizens are pretty much restricted to people with special education, knowledge, or experience. The medical/healthcare field may have the most demand. When I applied for a visa several years ago, the process took about 8 months even though it was just of transfer of the job I was already doing from the US to Italy. Your best bet might be to find a job with the office of a multinational company in the US and try to transfer from there. You're more likely to get relocation assistance that way too.
It's useful to check the expat sites for information about living and working here or other places you might be interested in:
http://www.escapeartist.com/
http://www.expatica.com/
http://www.expatsinitaly.com/
There are similar sites specific to just about any country you might be interested in that you can find by searching for "ex pat" or "expat" and the name of the country. Generally, these will have a lot of good information on daily life and negotiating the bureaucracy when you arrive and provide you with useful information you should know before you make the transition.
Right now, the economy isn't any particularly good and jobs are few and far between. The political climate is very much against people who are here illegally.