Best neighborhood to stay in in Lisbon?

You might want to look at Baixa or Chiado. Baixa is flat while Chiado is hilly. Both would have quiet areas but Baixa more of them. Public transport is good for both areas, but walking is also good. The Principe Real area is another option.

My wife and I just returned from Lisbon a week ago, after visiting the city for a few days in mid-April and another full day in early May. The Lisbon central district is hilly. Our first B&B accommodation was across the (4 lane) street from the Ritz hotel, which is on a high point, several blocks west and above the Marques de Pombol statue/major traffic roundabout/Metro stop. We walked down the relative steep hill from the B&B to the roundabout and continued southeast and downhill all along the Avenue da Liberdade. It was interesting and easy to walk to Rossio and beyond (all downhill) or up into Chiado. The Park Eduardo VII was just on the other side of the Ritz, and it also slopes, with wide walking paths, down to Marques de Pombol. Great views from the top. When we found ourselves down toward the river (estuary) and were ready to head back to the B&B, we would find the nearest Metro station and take the Metro back to Marques de Pombol station and hike back up to the B&B. Our second stay had us in a B&B in the Misericordia district next to the pocket park Placa das Flores. An interesting neighborhood that I would have liked to explore more. Again on a hillside, we walked uphill several blocks of small streets to access either the Rato Metro station or the No. 24 trolley, which apparently just began operations in early April. There is no paper map of the public transportation system available, but trolley & bus route maps can be found on the provider's (Cassis?) website. We each bought a 10 Euro zappable Via Viagem (?) card when we first arrived at the airport in April, used them on the Metro, buses, the trolley, and a ferry over 2+ full days (both visits), and did not use them up. We booked a shuttle to get us from our Misericordia B&B back to the airport for our 06:15 flight home. It cost more than the two Via Viagem cards, but I believe there is little or no public transportation in the very early morning.

Thanks! This really helps.

Bairro Alto.
AirBnB for us. 5+5 days.
This place https://www.airbnb.com/rooms/789789 in Lisboa was great!