I had a very similar question when I looked for a location for my husband and I to stay last October. We opted to stay in Chiado, close to the Rossio Train Station on the edge of the Barro Alto. After we visited each of the areas that you mentioned: Baixa, Chiado, and Alfama, we were glad that we chose this area, however, each of these areas have their pros/cons.
Baixa:
- Pros: Flat and easy walk with great access to the river, metro, trams, and train station. Many of the place you will want to visit are in the Baixa, easy access to sites.
- Cons: for me it was too touristy and it was more challenging to find authentic food choices. Shopping is also more geared to tourists. Most of the stunning views are located on the hills in Alfama or Chiado, not Baxia.
Alfama
- Pros: Vibrant neighborhood that is a pleasure to wander and get lost. Great street art, access to wonderful Fado restaurants, great views. There are really nice neighborhood restaurants in Alfama.
- Cons: Hilly and can be difficult to navigate. Access to transportation is a bit more complicated in Alfama than Baxia or Chiado. Not a place you would go for shopping. There are some cool boutique shops up near the Castle.
Chiado
- Pros: Nice blend between local and tourism. Great views, wonderful restaurants, good shopping options. Near the trams, metro, and train station (especially if you are going to Sintra). If you like partying and bars, you are a short hop from the Barro Alto.
- Cons: Hilly and a lot of up and down, especially if you find yourself spending a lot of time in the Alfama area. Can be touristy, but the public squares near the churches and the miradouro is so delightful to have a drink and soak in the atmosphere. Our miradouro was 5 minutes from our hotel and I went there every clear day to enjoy the views.