Largo do Carmo, Carmo Museum, Cervejaria Trindade, São Roque church, São Pedro de Alcântara viewpoint, Pavilhão Chinês, Principe Real Gardens, Embaixada, Coralla coffee and chocolate store, Entre Tanto Urban Market, Praça das Flores
You can do this walk at any time of day but given opening times of sights and venues on the route, afternoon is probably best unless it's baking hot. If you intend to visit the Carmo Museum, it is open from 10am to 6pm daily except Sundays. São Roque church is open from 10am to 6pm daily, except Mondays when it opens at 2pm. Embaixada and Entre Tanto open from noon to 8pm and Pavilhão Chinês is open from 6pm to 2am.
As in any city, you should keep an eye on your belongings as pickpockets tend to target tourists in crowded areas. You will also be going to some quieter spots but it's always best to take simple precautions.