"We are moving next to a beautiful city."
Translation:Egy szép város mellett haladunk.
Based on the vocabulary I'd seen up to this point I was able to guess what meaning was wanted, but the English sentence strongly suggests a meaning more like "Egy szép város mellé átköltözünk."
"Egy szép város mellé költözünk." or "Átköltözünk/elköltözünk egy szép város mellé" are correct. The version you wrote is incorrect. But basically you are right: "to move somewhere" means "elköltözni valahová".
I would interpret this English sentence more as "Next, we are moving to a beautiful city." To make it have the intended meaning here, I would reword it to "We are moving beside a beautiful city."
I hadn't even considered that possibility! Unfortunately, it depends largely on tone of voice (especially putting lots of emphasis on "next") to communicate this, and isn't immediately apparent to most just from the writing. :(
It doesn't matter if you say beside or next to. "Moving" is used here in a way that English speakers don't use it. If the implication is we're in a car, driving past a beautiful city, or in a train perhaps, you'd say: We are passing a beautiful city.
Ezt a mondatot bármelyik natív angol félreérti, soha sem használjuk ilyen értelemben a moving szót. Moving - költözés - passing by - elhaladunk mellette (szerintem). Esetleg traveling, jöhet még szóba.