It's principally right, but you wouldn't use "into" in that situation.
You'd use "into" if you stood just outside of the city and wanted to go in. In most modern cities you can't clearly define where a city begins, but let's say it has a wall as a border, then it'd be good. But in that case the Hungarian sentence would look a little different, too: "Bemész a városba?"
- Megyek a városba - I am going to the city. I may end up there, but also may stay in the outskirts cause there's a nice cafe there where I can spend the day.
- Bemegyek a városba - I go into the city. I'll end up somewhere in the city centre and do whatever I came for.
- Megyek a városba - I'm going to the city.
- Bemegyek a városba - I enter the city.
The -ban/-ben suffix means that you are on the inside of the object (English: in). The -ba/-be suffix indicates that you are moving from outside to inside (English: into).
To make the trinity complete, there's also -ból/-ből, which means that you're going from inside to outside (English: out of).