Take a train from Cascais to the end of the line at Cais do Sodré station in Lisbon, cost €2,15, takes 35 - 40 minutes. Trains every 20 - 30 minutes from 5am to 1am every day, more frequent during weekday peak hours.
Take the Metro from Cais do Sodré to the airport cost €1,45, takes about 30 minutes. Trains run every few minutes from 0630 to 0100 every day.
Total therefore €3,60 plus 50 cents for the rechargeable ticket card. so €4,10 I make it.
There are no buses and even if there were they would cost more.
I hope that helps.
For a family of five like yours if you do the math, the cost of a taxi will not be much higher and you'll not lose 1h30 to 2h00 doying it.