There is no shortage of English teachers; this would not be sufficient for a work visa.
The exact jobs depend on the country (eg for the UK, look on the occupation shortage list), but generally you'll need to be educated to masters level with at least 4 years of experience in a shortage field, often medical or technical. The employer can only sponsor your work visa once they've proved that they can't find an EU citizen to fill the role, and has to then pay for the work visa.
You'd also need to be completely fluent in the country's language; so unless you are, you'd be limited to the UK and Ireland.
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Europeans have no need for English language teachers from USA - In Europe, they do have English language teachers from UK and Ireland available to them without visa -
High education levels and qualifications are required to get hired within EU - It is very difficult to qualify for a visa in Europe - It is comparable to the difficulties that Europeans have to qualify for a visa to work in USA -
The only practical way to get a visa for employment in Europe, for an American is to be hired by a USA company and to be assigned for a position based in Europe -
Get hired by IBM and get based in Brussels, or by Delta Airlines and get based in Paris -
I would also suggest you better be fully fluent in a European language - Such as French, German, Italian, Spanish...
You do not go to a EU country consulate within the USA and ask for a visa - You FIRST get hired by company XYZ and company XYZ makes the visa request -
The town where I lived in Italy did have a language school where English was taught, but they would never hire someone from the US because there were a large number of candidates from Ireland and the UK who didn't need visas to live in Italy.
It is very difficult to move to a country in Europe if you are not European. You cannot just pick up and move to another country. If you are not European, you will have to arrange a visa in order to move there legally. European regulations require employers to advertise jobs and demonstrate that there are no viable European candidates for a job before they can offer a position to a non European. In the best of times non-Europeans have to have specialized skills, education and/or experience that cannot be easily duplicated by European job seekers. You're more likely to get a job that takes advantage of your education and skills than an entry level job; however, it's very difficult. Expecting a job when you don't already speak the local language or expecting employers to go to the time and expense to get you a work permit when that wouldn't be necessary for European candidates is unrealistic.
Right now, the overall unemployment rate in Italy is around 12.7% and it's around 42.7% for young people. In Greece, the unemployment rate is 26.5% overall and 56.9% for young people. In Spain, the unemployment rate is 25.3% overall and 53.9% for young people. In Portugal, the unemployment rate is 15.2% and 35.4%. In France, the unemployment rate is 10.4% overall and 23.4% for young people..
My direct experience is with Italy where I lived for the past 13+ years (working for a Swedish company); other countries here in Europe will have similar rules. It's not all that different anywhere else. The site for visas in Italy is: http://www.esteri.it/visti/index_eng.asp . The site has links to the application, the additional information you need to supply in order to get the visa and where to apply. It also includes education visas which are somewhat easier to get than work visas. You can find similar information for other countries on their consulate websites.Visa information on the website of the French Consulate in New York is here: http://www.consulfrance-newyork.org/-Vis... . You can find other consulate websites with a fairly simple search. Note that you'll need to apply to the consulate that has jurisdiction over the state where you live, but you can find the necessary information on any of the consulate websites.
A work permit is separate - you cannot apply for that yourself in many countries. 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 Canadian or US citizens are pretty much restricted to people with special education, knowledge, or experience ... and you would have to be able to speak the local language. However, the level of English is high in Scandinavia and most adults speak it very well. You would be able to manage well while learning the language. Right now, as noted above, the best bet would probably be something in the healthcare field which is a fit for you. When I moved here, it was through a transfer of the job I was already doing in the US to the Italian office. Even so, it took 8 months to put all the paperwork in place to apply for a visa. If you already have a job offer, the company will provide you the information you need for your visa application and take care of things like work permits.
It's useful to check the expat sites for information about living and working in the places you might be interested in:
http://www.expatsinitaly.com/
http://www.escapeartist.com/
http://www.expatica.com/
You can find other sites by searching for "expat" and the name of your target country. These sites will tell you how to register your address, provide information about healthcare, banks, and so forth to ease you into daily life.
You need to be aware that if you are a US expat, you would need to file tax returns in the US as well as in your host country and that can get to be expensive. You may also want to compare the cost of living in potential host companies to where you are now. http://www.numbeo.com/cost-of-living/
Best bet is to get sponsored by a foreign company, and also know their language well. You will be treated poorly if you can't speak the language.
