Your paragraph is very well written! Here is a step-by-step grammar check and explanation:

• First sentence: "Every part of the city feels different." - This sentence is correct. It's clear and simple.
• Second sentence: "The centre is noisy with cars and buses." - This is also correct. "Centre" can also be spelled "center" depending on where you live, but "centre" is fine.
• Third sentence: "The old town is pretty and full of visitors." - Perfect! The sentence has a clear subject and description.
• Fourth sentence: "In quiet residential areas, children play outside and mothers push strollers," - This is good, but it’s better to add "while" before the second action to show both actions happen at the same time: "...children play outside while mothers push strollers." Also, the sentence ends with a comma, so it needs to be connected properly to what follows.
• Fifth sentence: "and the amusement park is alive with joyful shouts." - This sentence starts with "and" because it continues the previous one. Instead, it’s clearer to combine it properly after the previous sentence. For example: "In quiet residential areas, children play outside while mothers push strollers, and the amusement park is alive with joyful shouts." Alternatively, make it two separate sentences.

Corrected paragraph:

Every part of the city feels different. The centre is noisy with cars and buses. The old town is pretty and full of visitors. In quiet residential areas, children play outside while mothers push strollers, and the amusement park is alive with joyful shouts.

If you want, you can make the last part two sentences for clarity:

In quiet residential areas, children play outside while mothers push strollers. The amusement park is alive with joyful shouts.