"My children like to walk around the city."
Translation:Мои дети любят гулять по городу.
Hmm... I always thought that гулять sounds more casual, as in "going for a stroll around the city", whereas ходить doesn't necessarily mean that you are going for a stroll. For example, "I go around the city taking pictures of different places." Please correct me if I'm wrong.
I've used and heard the use of ходить as a casual term as well. As far as I know, "ходить по городу" means a casual walk as well. Notice I'm using a phrase, not a single word. ходить literally means 'walk' anyway. Nevertheless, the usage of гулять here is correct. I'll ask a for native speaker's opinion.
гулять is explicitly "to stroll", "to take a walk". It works with по—which is important because you should not take for granted that ANY action can be done "all over" something.
ходить is just "to walk" in the sense of movement (on foot). Maybe you go for a walk, maybe you just wander around the city.