Yesterday, we went to the city CENTRE (...) but it BEGAN (...) to rain so we came back home.
Nie wiadomo czy autorka miala na mysl "city centre". Moze mieszka poza miastem?
"began to rain" znaczy to samo co "started to rain".
I try not to change a writer's words, particularly if they have written a phrase correctly. Your suggestion of replacing "came back" with "returned" was a good one. It interjected a little variety in the text.
And I agree that for the level of writing, replacing the word "unpleasant" with "bad" was a good suggestion.
"The weather was ugly" would be acceptable in informal speech in AmE, but I wouldn't recommend using it in one's writing.