In Lisbon, our family (2 kids, then ages 6 and 9) stayed in a great apartment on the edge of the Bairro Alto and Chiado districts. We felt it was the perfect location, as there was a trolley stop right on the plaza outside our front door (Praca Camoes), so we could easily travel throughout the city. The neighborhood was also within walking distance to many good restaurants. We found our apartment through www.travelingtolisbon.com</a>; the staff always provided prompt and detailed responses to any questions that I had before our trip.
We rented a car for greater ease of travel in Portugal, so I cannot speak to the train availability to nearby places. However, two of our favorite places (not along the coast, but near Lisbon, to the west) were the towns of Sintra and Obidos. I am not sure if your reference to "on the coast" refers to the western or southern coastal regions. For the western coast, we stayed a few days in the town of Nazare--it was nice, but not spectacular. We liked the energy much better in the adjacent town of Sitio (on a cliff beside Nazare, reachable by a funicular). On the southern coast, we absolutely adored the small village of Burgau--very charming, peaceful and "non-touristy". We rented a tiny apartment there, with a view of the sea. I have heard that you can get to Burgau from Lisbon by bus more easily than train.
We loved our travels through Portugal! I hope that you have a wonderful time there!