How to Move to Europe
So you want to move to Europe, eh? 10 years ago, I made the move myself, from the USA. Back then I was moving to a non-EU country, which then joined a few years after my arrival. Depending on where you want to move, the local customs and culture will vary, but the overall expat experience is quite similar. So is the process of becoming legal and getting permission to stay in the country beyond that of a tourist visa. Here are some things you'll want to consider before moving abroad, particularly if you're coming from the USA.
3rd Country Nationals
If you're not already an EU citizen, or married to someone who is, you will be classified as a 3rd Country National. Sounds kind of dodgy, and I have to admit that it feels pretty much the same. But that's what we are. Even if you do marry an EU citizen, you're still a 3rd Country National, just one who automatically gets to enjoy most of the same rights an EU citizen gets to enjoy. Mind, they are not your rights simply by way of marriage, they are the rights of your spouse, ie, it's their right to have you alongside them and living in the same capacity. Also, if you divorce an EU citizen, you run the risk of losing your EU residential status, but this depends on where you are living, and other personal circumstances.
Finding a Job
Moving from the US to the EU (or a non-EU country in Europe) is not as easy as many seem to think. The older EU countries have very strict rules about immigration, and if you are traveling solo as a US citizen you will need to have a job lined up before you get here, and in many cases you will already need to have your work visa in hand before entering the country. If you're highly skilled, many EU countries have a program for this and you stand a better chance than most of finding a job and having a company help you get set up. If you are going to navigate the red tape by yourself, no matter which country it is, you had better have loads of time, patience and money at your disposal, because it can be a trying experience.
Getting a Visa
There are several types of visas available and different countries may have slightly different procedures. There are work visas, which usually require finding a job offer first and having the company walk you through the red tape, and there are business visas where you could establish a company or get some sort of entrepreneur's license and get legal that way. (There are other types of visas, such as family reunification, etc, but most people aren't able to go that route, so I'm not going to waste time discussing them here.)
Business Visas
I've already mentioned the work permit and work visa method, so let me focus a bit on the business visas. Many EU countries will let you set up a company, but this requires a boat load of money, not only to pay someone to get everything up and running, but also the initial investment. You may also be required to hire locals, etc. If you are independently wealthy, this could be a great option for you -- but for most of us, this is a massive expenditure that cannot be afforded. Then there's the entrepreneurial method: You can set up as an independent contractor, but in some cases this requires a huge investment as well. And, some countries require that you first be legal on other grounds for 2 years before being allowed to set up a trade license. You'll need to research a particular countries rules and regulations well before you try any of these things, and while it can be costly, it's often best to hire a professional company that does this sort of thing for people -- just be sure it's a legit company before you hire anyone to help you.
Accomodation
Believe it or not, many European countries require that you first have a signed lease for accommodation before they will give you the visa, even if you've already fulfilled the other requirements. The lease will need to cover the length of the visa (usually, anyway) and this could be difficult if you don't have time to go flat hunting in the foreign country before your arrival. In some countries, like the one I live in, finding a flat is very easy to do, but you're required to pay 1st months rent, last month's rent, and realtor fee in the amount of 1 month's rent as well. This makes getting a lease quite an investment if you're not over here yet, and if you've never even seen the flat. And in some places, like Paris, getting a long term lease can really, really be difficult because landlords are very careful and particular about the terms of the lease, since it's quite difficult to kick bad tenants out. But I think France is an exception to the norm...
Documents
To move to Europe and get a legitimate job you will need a number of documents. Your passport will need to be valid for a good length of time, usually at least the duration of your intended stay so that it covers the length of your visa. In some cases, it must be valid for 2 years beyond that. In many countries, your birth certificate and things of that nature cannot be more than 6 months old, and in some cases not more than 3. This means you need to get a new copy of yours issued before you get here -- don't worry, that's easy to do, just contact the vital records dept of the state you were born in. You should also have any documents like this super-legalized with an Apostille before getting here, cos it's very hard to do from overseas. It basically validates your document for use in Europe and elsewhere. You should have one for your birth certificate, marriage certificate or divorce decree, etc. They need to be issued by the Secretary of State of the state that issued the document. Just look up the website and follow the instructions, it should be clear enough.
Bank Accounts
You should not come to Europe without some sort of savings. Things cost money and you never know how long you might have to wait for that first paycheck. You can use a US account and withdraw from an ATM, as they are everywhere over here, just like in the US. You will probably need to open an account in your new country as well, just do yourself a favor and find a branch that has English speaking associates, unless you happen to speak the local lingo already.
Renewing Visas/Work Permits
If you're on a visa, you're going to need to renew that at some point. This is often expensive if you're not able to navigate the red tape on your own. Some countries only issue a 6 month visa to new applicants, which means you pretty much have to start the renewal process shortly after getting it. Other countries issue first time visas for a year. Most will not issue them for more than two years. In some cases the terminology will vary. A visa is actually usually something that tourists from non-visa-waiver countries need (Americans typically don't require visas if the stay in the EU is less than 90 days, but after that 90 is up, you need to outside of the EU for another 90 before you can re-enter it.) If you're given a year long "visa" it's probably actually a "long term resident permit" but many office clerks will still call these visas because that's what they used to called, regardless.
Permanent Residence
If you've been living (LEGALLY) in an EU country for 5 consecutive years, you can apply for what's called "permanent residence". This gives you nearly all the same rights as an EU citizen. Once you have this, you'll get an ID that needs to be renewed every 10 years or so. If you leave the EU before those 5 years go by, you have to start all over, so if you've already been there awhile, it's probably worth it to hang on for the permanent residence, because then you can also be granted EU Long Term Residence, which basically means you can then move anywhere in the EU that has a reciprocal agreement with the country you live in, which in most cases, is all of them save 3 or 4.