The visa is not the problem. As a US citizen, you generally do not need a visa to enter a European country. Many/most of the times (depending on country) your passport is not stamped. And you can legally stay for 6 months. My wife (US citizen) and I (still German citizen) moved to Germany for 2 years, and in the first year, we didn’t bother with any papers for her, because she was travelling in & out anyway (resulting in 6 months every time), and if we would have been sticklers for rules, a quick trip to Belgium or Holland (30 minutes away) and back would have resulted in another 6 months. Which are not tracked.
They don’t need to track at the border, because everything is tracked inside the country. Work, taxes, social security, health insurance, every little move builds a huge database. In my wife’s case, it didn’t matter, and they didn’t bother, because she didn’t work. But I do not recommend to work without at least applying for the proper papers. If my wife would had been here by herself, she wouldn’t even have found a place to live. In Germany, you need to “register” wherever you move, another set of fields in the database, another requirement for actually existing here.
That’s why you need your work permit and temporary residence permit.
The enter-as-tourist gambit is as popular here as it is in the US. But I would get the application process going ASAP. To get paid, you need a tax payer ID, no papers, no ID, no money. An understanding employer may pay you in the US for a few months, but as Penny said, some HR departments work slower than the Government (and, I may add, work more according to the book than some government agencies.)
Invisible: If you are married to a Swedish citizen, the residency is a no-brainer. The work permit must be applied for separately. Go, find a job, and then apply. Most EU countries treat the US wives of their citizens with very well. Now, if you would be from Turkey … (Reason for the non-automatic work permit: Unemployment benefits are usually generous, and they want to keep you out of the system as long as possible. Once you’ve found a job and got you work permit - in that order - you can do what many EU citizens do: Get fired and collect unemployment benefits …)
Personal recommendation: Watch your driver’s license status. An international driver’s license doesn’t mean a thing, it’s simply a translation. You typically can use your foreign license for a year, and you can only apply for a local license after half a year, supposedly to gain local experience. (may vary and change according to European standardization…) You do not surrender your US license and you get a local one. According to international agreements, they have to give you one. Feel good about it: The guy in the car next to you had to study long for his license, flunked the test maybe 3 times, and spent more than $1000 for the whole thing. (Except in Belgium, where, according to European lore, you win it in the lottery. It’s not true.) Hold on to your license when you leave: When we left, the German driver’s license was good for life.