Welcome to Europe! I hope you'll have fun over here. Lots of things can or will be different, but not necessarily in a bad way.
Laws are different. Sounds kind of obvious, until you learn that some countries have serious sentences for things like littering or chewing gum in public. Read up on the basics just in case. Knowing things like the fact that several EU countries demand that anyone can show ID at all times, so carrying an ID card or passport is mandatory, can save you from an awkward fine. Your foreign drivers license is NOT valid ID!
The legal system may work differently. The "you have the right to remain silent" thing may not apply. Lawyers may work differently. The country you're relocating to may not have juries as part of the legal system. You're probably not going to need that information, but you're better off reading up on the basics just in case, you don't want to get surprised.
Police violence and racism still exist in Europe, though it usually manifests in a different way. The difference will probably depend on your background and where exactly you're coming from, but there's no direct need to be nervous around police officers. Most of them don't have guns, and if they do, pulling them out at something as banal as a traffic stop will get them in trouble. However, if you're awfully nervous for no reason, they may start to get suspicious.
Some countries offer some social benefits to only their citizens, not to legal migrants. Make sure you know what you are and aren't entitled to so you don't find yourself footing a massive medical bill. Also check for any mandatory insurances that may not apply to you now.
Not every American credit card is widely accepted. They'll work in things like hotels, but maybe not for parking infrastructure. In some countries, credit cards barely work in grocery stores, for instance, especially in debit card-first countries. Taking out money abroad also costs quite a lot, so look into if you need to get a local banking account.
Your driver's license is likely valid, but needs to be converted to an international driver's license for it to be recognized as such.
If you get a mobile phone subscription from an EU country, mobile roaming charges are basically non-existent. However, if you're traveling, make sure you remember what countries are EU or not. Unexpected Swiss data charges from driving along the border can ruin your week.
If you're between 18 and 21: the drinking age is lower. Please don't drink yourself into a coma. Additional alcohol laws may apply: some countries only allow selling alcohol at government run facilities, for instance.
Smoking rules and practices differ per country and region. Some countries ban smoking in public outright, others still very much smoke everywhere.
Some things will be a lot cheaper (healthcare), others will be a lot more expensive (alcohol, cigarettes, fuel). If you've planned ahead on your finances, make sure to use local prices.
Many American brands have identical packaging for differently flavored products. We have coke and Fanta and Pepsi, but they may not taste like you expect. Same goes for a lot of food: many foodstuffs are still filled with sugar, but from what I've heard American food is even more sugary somehow, so food may taste a bit bland when you first get here. If you have a healthy diet, that should correct itself in a few weeks.
I've never seen those mythical free soda refills mentioned in American movies in Europe. An expensive four euro glass of soda probably doesn't come with a free refill.
Housing may work differently. Every country has their own rules for social housing and who can apply under what conditions. There are huge businesses out there that make money off scamming rich expats because of this. Make sure you're informed of the local housing situation ahead of moving, unless your company arranges (temporary) living arrangements for you. Some expats in cities like Amsterdam find themselves struggling for months to get a basic apartment, wasting a lot of money on hotels despite their lavish income.
I'd recommend learning the local language, even if everyone you interact with at work will speak English. Quite a lot of expats find themselves socially isolated because the local populace doesn't want to speak English when hanging out with friends. Europeans, especially the younger half, generally understand English to the point where you can make yourself clear, so you have time, but if you want to make friends, you need to integrate.
As you've probably heard, tipping is different. Tipping differs from "optional, but very welcome" to "insulting because of associations with bribes" depending on where you go. Read up on local customs if you're not going out with locals.
Advertised pricing is almost always including tax. Don't add the 19-21% VAT in your head like you would with some American taxes, expect to pay the price as advertised. There are some exceptions (bottle deposits, for instance, but you can get that money back by handing in bottles at the right places) which will differ per country, but those shouldn't break the bank.
Some countries are moving towards becoming cashless in practice. I use cash about five times a year, plus about a third of the times I visit Germany. Some stores may not accept cash. If you combine that with the "not all stores accept American credit cards", you may find yourself in a pickle. Luckily, things like Google Pay and Apple Pay use a universal protocol of sorts, so you can try setting that up as a backup.
Check out how public transport works. Some countries let you travel for free, others will require paper tickets, and there are countries where your credit card can act like a travel pass. Some places sell tickets inside trains, others will demand you buy a ticket before entering.
Traffic will work differently. Traffic signs are usually quite universal (stop signs even say STOP in France, unlike in French Canada!) but there are additional traffic indicators that probably work differently. Think painted lines on the ground to indicate right of way, special asphalt colours to indicate bicycle paths, but also traffic light placement. A quick read up before venturing into traffic would be wise. Some countries have a system where traffic coming from the right in an unmarked, unprioritised intersection will have right of way. If you're driving or taking a bike, better check up on when you need to stop! Also, crossing the road is generally not illegal ("jaywalking"), but use common sense when to use that freedom.
Lastly, the tax thing: unless you earn serious bank, you're probably not paying extra taxes in the US. However, you are obligated to fill out tax forms. How the tax situation works depends on the country you move to. Some will have you pay American taxes and subtract them from local taxes, others will work the other way around. Either way, you're probably not paying extra; despite what some people claim, tax pressure within the EU and America is actually quite similar. You will end up with extra paperwork, though. Speaking of, check the pension situation for the job you're working for. If you don't check, you may end up paying into a foreign pension plan you won't have easy access to when you retire.
Freedom of speech exists across the EU. However, it's not the rather absolutist freedom of speech Americans (think they) have. If you pull an Elon Musk hand wave in Germany, you may end up in a cell. Libel laws work differently, insults may work differently. You may not have the right to record police officers. Hate speech is generally not protected. Some countries have laws against blasphemy. If you take part in any protests, make sure you know what you can and can't say in public!
Everything will be measured in metric. Kilometers, kilograms, liters. Some countries have local language equivalents for things like pounds or ounces, but they don't line up with your pounds or ounces. You'll hopefully pick up real quick that a meter is about three feet and a yard is about 100m and such.