My first trip to Portugal was at the end of April and I loved it. The weather was pleasant..just warm enough. Portugal is a gem that not nearly enough Americans visit. After a few days in Lisbon, we went north to Sintra, Alcobaca, Batalha, Nazare, Obidos (OK it's touristy but nice),Viseu, Coimbra. Near Lisbon, be sure to visit Queluz( a little gem of a royal palace), Estoril and Cascais.
Don't miss the Gulbenkian Museumin Lisbon which is really incredible. It is easily accessible by subway and a long walk, as I remember.
On another trip, we stayed at the Pousada in Evora, which was truly wonderful. Evora is very interesting.
You usually have to reserve far in advance for Pousadas which are also not the cheapest place but worth it.
Enjoy your trip and do a little research on Portuguese history in advance.