1 - Las Ramblas: take a bike or nice sport shoes and cycle/walk part or every 25 kmof them looking at the city and the "sea".
2 - El "Mercado del Puerto": you should go on Saturdays but anyway, it is a nice pavillion with some overpriced and some cheap restaurants with all the Uruguayan meat you could always want.
3 - Pocitos: Like Palermo, but in Montevideo. Nice streets.
4 - The center, run down and melancholic; the National Assembly, the sister building of the Argentinian Congress. Also, Parque Rodó.
5 - About the "cerro", Ive never been there, but supposedly is nice, but in a bad neighborhood, any help with this one?
6 - Take a bus to Punta del Este for a couple of days, or Piriapolis if you like quiet, sea and hills, or head farther if you are staying more (Valizas, Cabo Polonio, Punta del Diablo).
7 - One hour from MVD there is a town called Minas, is small and have small hills around it. It is green,gentle, and great to walk and see real small town uruguayan life.
8 - If you can get to a "murga" show (the main thing is in February, but maybe there is a retrospective or something) go to see it.It is the special uruguayan carnaval thing, generally groups of twenty singers at the same time with satirical songs about the country and politics. Google "Agarrate Catalina" (one of the main groups) and see if you like it.
Remember, MVD is not BA.It is like a time machine, like going back 10 or 20 years. It is mostly safe but it needs a good painting job. And I almost forgot: Drink LOTS of mate, lots of it, the locals will help you with that. And listen to a tango while at it.