The best thing you can do in any new neighborhood is also the simplest: leave your hotel in the morning with no plan at all, and walk until you are hungry. No monuments. No museums. No 'must-see' list.
This is the technique the French call flânerie, and it is the only reliable way to actually understand a neighborhood. You will see things you would never see on a guided walk: how locals greet each other, where the school run happens, which corners are noisy at lunch, which streets are deserted at 3 PM, where the old men play chess.
The mistake most travelers make is trying to optimize. They plan routes, they tick off landmarks, they chase the perfect photo spot. All of which produces the same trip everyone else has. The aimless walker, by contrast, leaves with a story nobody else has.
Our suggestion: dedicate the first morning of every trip to a walk with no destination. You will thank us for the rest of your stay.