Where to stay outside of Lisbon

Sintra & Cascais are very easy day trips from Lisbon by train while Lagos is almost 300km south. You would need 2 home bases to include the Algarve area.

Yes, both Sintra and Cascais are easy day trips from Lisbon by train - that's how I saw each of them in May. Staying in Lisbon worked well for me, but people have said they enjoyed spending a night or two in either Sintra or Cascais, and I can see why. Sintra especially is a relaxing little town that is very touristy during the day but much quieter at night. Only recently Sintra made some of their streets pedestrian-only which might make driving there less fun but easier for tourists coming by train. I'd avoid staying in Lisbon only if you dislike cities, however, and add nights in Sintra and/or Cascais only if you have extra time. Some of this is just a personal preference; many would prefer to spend all of their nights in Lisbon and do the other towns as day trips.

With only a week, I would skip Lagos and stay in Lisbon or close by. You can definitely fill a week in Lisbon with day trips to your mentioned towns and maybe add Evora as another day trip.

Sintra has lots to see, but I have become very disenchanted due to the high number of tourists and the city’s attempts to improve things by making the downtown streets pedestrian only. It has added to the chaos because they haven’t done a good job of communicating new bus routes, etc. I would recommend a small group tour to Sintra from Lisbon or a private one.
And, you don’t need or want a car if you stay in either Lisbon, Sintra or Cascais. Parking is difficult and expensive and the public transport is great - with the exception of buses in Sintra..