We have seven full days in Portugal the last week of March with a 13-year-old. We typically like to do sightseeing or an "activity" each day while staying at a lovely hotel. If we start with Lisbon, where should we spend the second part of our trip? The Algarve? Azores? Medeira? Smaller towns? Will most likely stay at the 4 seasons there. Welcome other hotel and dining and activity suggestions!
For the second part of your trip I highly recommend that you visit and stay in Porto.
I just returned from a week-long family vacation to Lisbon, Sintra, Cascais. My wife and kids were complaining that they wished they had more time! While in Lisbon, we made a day trip to Sintra. I found it a bit of a rush and perhaps an overnight stay at Sintra might work better. There are lots to visit at Sintra but the key ones are Pena Palace, Moorish Castle and Quinta da Regaleira. We ended up not having enough time to visit Quinta on a day trip because of the crowd (even in December when it's supposed to be a low season).
We stayed at Iberostar Selection Lisboa which is a bit further north of the centre because the hotel offers a very good buffet breakfast (lots of good food & great service). The idea was to save time by having a full breakfast in the hotel before we went sightseeing and not have to rush had we gone out for breakfast at a cafe or restaurant; we found Portuguese restaurants in general to be quite slow in their service, maybe that's a cultural thing because everyone seemed laid back. But we didn't want to spend an hour or more just eating breakfast because we were eager to go out for our activity every day, and so the hotel buffet breakfast was great for us.