Shanty tour? You're referring to what are known as "pueblos jóvenes" ("young towns", literally) in Lima, slums that extend for many miles in the outskirts of the city. They are not particularly interesting and can be downright dangerous especially at night, although I suppose that the idea of a tour is to manage your security risk. In any event they are not museums. Leave those poor people alone, please. It makes me gag to think of a bunch of tourists taking snapshots from a tour bus in those desperate places.
Culinary tour? I'm sure these exist, but I'd advise against a tour as they are likely to take you to places like the Costa Verde and La Rosa Nautica that are frequented mostly by tourists.
The best culinary experience (IMHO) in Lima is the seafood lunch in one of the many good "cebicherías" in Lima. Many posters including me have listed some of the better restaurants; do a search in the Peru forum. Briefly, ask around for Mercado, La Mar, Segundo Muelle, Pescados Capitales, Alfresco, Los Cavenecia, Kapallaq. You'll get the real thing at any of these places. None of them are located anywhere near the Sheraton, by the way.
Seafood, especially cebiche, is strictly a lunch-time thing in Lima but there are oodles of good places for dinner: Astrid y Gastón, Central (if your wallet is fat), Rafael and La Gloria are a few that come to mind. If you're into pizza a good choice is the Antica Pizzeria (there are at least three outlets in Lima) and for something less formal in a café/deli setting try San Antonio; these are scattered throughout the city.