The Rambla: This waterside roadway has people biking, fishing, drinking mate, and enjoying the great views. 22 kilometers-long (13.6 miles), the Rambla goes along Montevideo's waterfront. Lovely at sunset.
La Feria Tristán Narvaja Flea Market: Spend part of Sunday morning with the locals on Tristán Narvaja Street, where vendors sell everything from t-shirts to antiques to kitchen supplies. It's right off of 18 de Julio Ave. and the entrance is often marked by people selling puppies.
Pocitos: This barrio lies about 2 miles south-east of El Centro. The Pocitos beach runs east from Punta Trouville for about a mile. Highrise apartments ring the beach along the Rambla, but going in-land a few blocks brings you into an older neighborhood reminiscent of San Francisco's Marina district. Head uphill on 21 de septiembre St. from the Rambla at Punta Trouville for about 7 or 8 blocks to avenue Ellauri, turn left and walk another 4 blocks to Punta Carretas Shopping, a major shopping mall that is built on the remains of a prison (they preserved the prison gate inside the mall).
Walking: Perhaps not an especially beautiful city, Montevideo is a relatively safe one. The city is built on a slight hill, the spine of which extends into the Rio de la Plata to create the point that was the original city (Ciudad Vieja). From the Plaza de la Independencia, the main street that extends east from the plaza is 18 de Julio Ave. El Centro (downtown) is in this area and there will be lots of shops and places to change money. You can walk around without worry almost anywhere, and there are lots of side streets and areas you can explore: be aware that the port area, just off the main tourist and port terminal areas, is considered dangerous by locals as much as by the police. Parts of the city may appear run-down, but do not confuse this with it being a bad neighborhood. Along with Buenos Aires, this is one of the few cities in South America where poverty is not overly prevalent. That being said, there is simply not enough money in Uruguay to construct lots of new, modern buildings, so buildings are kept in use for long periods of time.