Chiado and Alfama are good areas, yes. They are very close to transport and to the centre, for shops and views.
Definitely go to Belém (day trip is fine) and Sintra (you may need two days to see it all), where you can relax a bit between site visits.
If you are staying for a week, take the train to Cascais. You get a nice view of the river/coast, and then can relax around Cascais, even on the beach, weather permitting.
Also try Parque das Nações, towards the North-East. There's a shopping centre, many restaurants, and a big aquarium (Oceanario), if you like that sort of thing.
Avoid Bairro Alto to stay, as that's mainly night life and can become noisy.
You can check Parque Eduardo VII (Edward VII), which has a huge greenhouse (Estufa Fria, or "Cold Greenhouse") - great for nature if it rains, and then Gulbenkian Foundation, with a lovely park with lots of art and a couple of Museums.
If you like woodland walks, try Monsanto.
It's easy to travel by bus, tram or train. In the city a taxi can cost as low as 5 euros, if transport is more complicated, and there's Uber too.