I would recommend Sintra -- take your kids to the Castle of the Moors:
https://www.sintra-portugal.com/Attractions/Castle-Moors-Sintra.html
and maybe Quinta de Regalaria:
http://www.regaleira.pt/en.
From Lisbon you can take a train to Sintra and then there is tourist bus loop that takes you to all the sights. Be warned it is a popular destination and gets very hot and crowded. Go as early as you can and hit the Castle of the Moors first, which is a nice hike. It's super cool.
Same thing with the castle in Lisbon, It's really fun, you can see peacocks roosting in the tree, and it is very kid friendly.
https://lisbonlisboaportugal.com/Alfama-Lisbon/lisbon-castle-st-george.html.
Plan your outings early in the day, rest during the midday heat, and go out again the in the evenings.
You don't need a car; the subway is great and taxis are very cheap.