The Bairro Alto in Lisbon, without being as old as the Alfama, is the most pure and picturesque place in the city. It's between downtown Lisbon and the front of the Alfama. It's a neighborhood with a lot of activity, (especially after dark) that you should see on foot, see its shops and breathe its essence. You should definitely see the famous Convento do Carmo neighborhood even though this is not a technological place. It's a residential and commercial area where you find the coolest fashion stores as well as the more "in" bars and restaurants in Lisbon. What's hot right now is there. To find something similar to Barrio Alto in Madrid, the equivalent would be La Latina with regard to the nightlife and Chueca in respect to the trendy restaurants and stores that are there.
Graça is one of the most picturesque neighborhoods of Lisbon. Walking through it is like stepping back in time and discovering unique places and stores. It's ideal for either walking, despite the number of slopes throughout the streets. No doubt, to discover the essence of Lisbon there's nothing better than spending a day walking through the streets, surrounded by the people. The tram crosses the entire neighborhood so it's easy to use.