Search for a job before you leave for your European country of choice. When you enter the EU on a travel visa, which for American citizens is a period of 90 days, you are not allowed to use that time to search for employment. Although this is a technicality that may be difficult to prove, some countries, such as Germany, require you have to leave the country with your tourist visa and then re-enter with a work visa. Finding a job before you enter the EU will save you time and money.
Wait the required workforce demand time. In most EU countries, a company wishing to hire a non-EU citizen must first post the job to the general EU public and wait a specified amount of days before they can officially start the process of offering the job to a non-EU citizen. You cannot apply for a work permit until that time has passed.
While you are still in the United States, go to the consulate for the country where you wish to work. For example, if you have found a job in Budapest and a company willing to sponsor you, you will need to go to the Hungarian Consulate--located in Washington DC--to apply for a work visa. Some consulates may accept applications through the mail, although this is at the discretion of the individual countries. In most cases, you will have to apply in person.
Apply for residency. Many times countries, such as Germany, require you to have a residence visa in addition to your work visa. This usually will state that you are allowed to live in the country for as long as a particular company employs you. When you discontinue your job with that company, you will most likely have to leave the country.
Wait for approval from the country's immigration office. Once you have been granted a work permit, you may enter the country for the time defined on your work visa.
After moving to Europe, attend all required meetings with local immigration officials and agencies. Failure to attend meetings or to report to the immigration offices if asked could result in a revocation of your work permit.
