To immerse yourself in azulejos, it’s best to visit the Azulejos museum in the Convento de Madre Deus and buy their guide. After this, the main monuments to see azulejos are: The Palácio Nacional in Sintra; the São Vicente interior and cloisters (esp. the La Fontaine fables); the Palácio dos Marqueses da Fronteira (nr. Monsanto park); the Queluz Palace; and the Igreja de Nossa Senhora dos Cardais (nr. Bairro Alto).

The following are beautiful — and also free to visit – so wander in from time to time!:
– Igreja de São Roque (look for the St Roch’s dog in the 16th c. tiles in the chapel on the right), in particular the tiles by the master Francisco de Matos
– Cervejaria da Trinidade (19th century, but colourful)
– The convent of São Pedro (behind the miradouro): you’ll have to ask one of the nuns nicely to let you see the chapel.
– The officers’ mess on Campo das Cebolas (alongside the market building at Feira da Ladra): you can walk inside and ask the receptionist to let you look at the fantastic tiles along the staircase, including plenty of ‘Figuras de convite’.

Further afield:
– The Ermida de Santo Amaro in Alcántara
– The Capela de Sao Filipe inside the Pousada at Setúbal (a great place for tea!).
– The Quinta dos Azulejos in Lumiar (difficult to arrange visits, but worth it).