I agree with cubsur's comments.
There used to be a bus operated by Scotturb but this was suspended last year owing to poor traffic loads.
With all your luggage, kids etc you really do not want to be struggling to get a bus to Cais de Sodre, buy tickets and then from Cascais station to your destination. Although it will work and I would probably use it if I was on my own, (the train trip has great views ), with the logistics I would recommend one of the following three options:
1) Taxi - Lisbon - Airport to Cascais
To arrange a taxi, check with the Tourism Desk in the Arrivals Hall at the airport when you arrive. I would think you would be looking at +-€ 60, although prices rise at night and week-ends and baggage is extra.
2) Private Transfer
Low Cost Transfers are currently quoting € 56 for a March transfer Lisbon Airport to Cascais.
See:
http://www.low-cost-transfers.com/lisbon-airport/index.php
3) Hire Car
Some rental operators in Lisbon have daily car hire offers. Unfortunately, I can remember who, as it a year or so I saw the adverts when up in Lisbon. I don't normally recommend hiring cars if it is one's first visit to Lisbon/Cascais but it may be an option for you.
