What are some of the things someone permanently relocating away from the US should be aware of?

it's much easier than you think it's going to be and is practically all upside.
I've been an expat for a decade and a half.
the other commenters seem to be offering incorrect theories, and I'll fact-check those
If you have a company offering you a job, they'll be sponsoring your work visa.
If you want to open a bank account in europe, you'll be able to with your work visa.
you will not be "on the hook" for US taxes if you are permanently relocating, living outside of the US for more than 330 days per year means that you still have to file US tax paperwork, but you don't have to pay $125,000 USD of earned income per year.
as for American chocolate? its maybe the worst chocolate in the world. Americans have terrible chocolate.
American chocolate is to real chocolate what American cheese is to real cheese. an ersatz barnyard embarrassment.
there are no American goods you'll want that you can't get or find a substitute for abroad, Im half convinced that comment is satire for using American chocolate as an example.
If you have any specific questions, go ahead and ask.
Good luck! whatever the deal is, moving out of that country is the right thing to do right now.
- If I have a remote job in the US but also have dual citizenship with the country I'm moving to do I still have to pay taxes on that income or does my employer have to do anything for my move.

If you go somewhere where the main language isn't English, you should make an effort to learn the local language. Yes, there are places like the Netherlands where virtually everybody speaks English well, but not putting in the effort means you'll always be an outsider.
Things are smaller outside the US: vehicles, apartments, kitchens, refrigerators, etc. It can take a while to get used to not having the same space you're used to. Also, some devices like clothes driers are rare.
Europeans take recycling seriously. In some places you have to pay to throw away garbage, while recycling is free. But, recycling is sometimes a real effort, like there's not a "glass and plastics" box, you have to take the clear glass to one place, the brown glass to another, etc.
Tipping mostly doesn't exist. That means that if you go to a restaurant you don't normally have one assigned waiter. Whoever is free will help you, which tends to speed things up a lot. OTOH, since they're not working for tips, the waitstaff don't feel the need to fake a smile, pretend to be your friend, etc. Some Americans think that comes off as unfriendly.
Electricity is more expensive (part of the reason for the smaller appliances) so sometimes will completely unplug things that an American would just turn off (like a TV).
Businesses don't have the same convenient hours as in the US. In some places, like Switzerland, they almost completely shut down on Sunday. 24 hour places are much more rare.
The European take on freedom of speech is different. You are simply not allowed to say certain things. Some things, like libel laws, are much more friendly to the person who is the target, rather than the "free speaker".
Oh, and smoking is still much more common in Europe, and it can be pretty disgusting. In the US it has been largely eliminated from public spaces, and smokers are confined to small smoking areas. Even in private homes people will often smoke outdoors either to be considerate or because their landlord doesn't allow smoking indoors. In Europe, smoking is still common indoors in many places, and... ugh.

You'll still be on the hook for US taxes.
- If you plan to return
  
  Even if you never plan to return, you are still (legally) on the hook to file a 1040-NR form with the IRS every year.
  There's a foreign earned income tax credit: this reduces your US tax bill by any income tax you paid to your residence country. For many expat working stiffs, this means they don't have to pay anything to Uncle Sam, but they still have to file a tax return.
  
  Even if you plan to renounce your citizenship, that's a long (and often expensive) process and you have to keep filing and paying taxes until it's done.
- The US is one of only 2 (?) countries in the world that does citizenship-based taxation.
  
  Is there an exit tax when you leave the US? France for example has that, and the new Belgian government wants to introduce it as well.

Opening bank accounts can be difficult due to the reporting/spying requirements banks have on US citizens. Actually not just citizens , I think it is called US persons of interest or something like that. Anyway not all banks want to deal with that and will not allow you to open an account.
- FATCA. You can usually search based on that.

There are languages other than English here
- That's no big deal. As long as you KEEP. REPEATING. YOURSELF. SLOWLY. AND. REALLY. LOUDLY, they'll figure it out.

Unless you formally revoke your citizenship you'll be required to file for taxes for the rest of your life.

You need a work visa. You can't just accept a job offer and move. The company would have to sponsor you and there's paperwork involved to allow you to legally work in that country. Some countries have limits to the type of jobs they allow foreign workers to take, and it's usually the ones they are lacking labor locally. There might also be minimum qualifications (i.e. education level, language skills).

There may not be an equivalent to Miranda Rights, namely a right to remain silent or guaranteed explanation of what rights you do have in the event of an arrest or other circumstances.
IIRC the UK can and will hold someone indefinitely if they refuse to talk. Which wouldn't be as problematic if they could stop abusing anti-terrorism laws, particularly against protestors.

As US citizen you're on the hook for US income tax. No matter where you live
- Cute of you to assume plebs leaving US will be making 100k or whatever threshold it is now...
  There are very few places that have similar salaries and they don't accept plebs.

This depends on your destination, but you may need to unlearn the habit in tipping, as wait staff salaries are reasonable. Where I live it is common to enable the option for various services, but you're never really expected to. I might tip if something was far beyond expectations with staff going above and beyond what is expected of them, but I usually don't.
Extremely simplified/rounded metric: 1 meter = 1 yard or 3 feet
1Kg = 2 pounds
1 gallon = 4 liters (a standard serving of beer is 0.5 liters. Sometimes 0.33)
Almost everything carries some sort of tax. This is usually included in the price tag.
50mph = 80kmh
most intersections don't have a stop sign. Yeld for anyone entering from your right hand side (unless you're in the UK or Ireland, where it's opposite)
Roundabouts are very common, but simple to learn. Think of it as a circular one-way street where you yeld for anyone already in it.
Learn the difference between Europe, EU, EEA and Schengen Area. In most cases you may be safe presuming they're the same, but there's a difference that may be relevant if traveling between European countries.
Most people speak English to varying degrees, but learning the local language is still very useful. Unless you're going to Denmark. Nobody understands Danish.
candy can be very different. European wine gum tastes very different from its US counterpart. The European version is of course far superior, and that's an objective and unbiased fact.

Which language do you speak? While many employer, especially in qualified field not working with public may accommodate English speakers, it's not as universal as you may think.
In qualified job, expect a loss of salary, well a part of it is covered by healthcare, daycare, school and more but at least at my work rk US laboar is more expensive when budgeting projects
You're an immigrant, expect to deal with all the associated bullshit
It's not like we don't have our share of far right politicians who hate everyone including themselve

you cant own a gun easily. no one just carries guns around either
healthcare wont bankrupt you
- Less guns and more healthcare? What a bunch of weirdos!
  
  Oh, there are a lot of guns. People just don't carry them around.

When you get homesick and want to eat some American chocolate or whatever, you won’t find it just anywhere. You’ll have to go a special American store for that. You know, like one of those Asian markets, but for American stuff. Everything in there is also really expensive, so don’t expect to visit every day.
- Honestly, since being introduced to the other side, I probably would never care to have it again anyway. It just tastes so much better to me. Give me a mars bar or Maltesers anyday over milkway or whoppers.

In general, the country in question should have a government (approved) website giving some information on what one should be aware of if planning on living there for a while or permanently.
There is an EU related website (don't remember which) where one can be provided legal counsel in regards to their right in EU specifically. You might want to search for it in the Europe oriented communities.
Don't expect the magical land of Oz. Each country developed in a different way, but we're all tinged by the darker elements common to the world at large.
Scalpers will try to take advantage of you. They love naive foreigners.

You can't bring your gun