The preferred alternative option for travelling from Lisbon Airport to Sintra is by Metro/Rail. There are +- 80 service per day taking about 50 minutes.
From the Airport take the Metro Red Line, (Linha Vermelha) to Gare Oriente (Metro/Rail/Bus Interchange). - 3 stops.
Proceed to the mezzanine floor where you will find the www.cp.pt(rail) ticket booths.
Buy a ticket and proceed up to the relevant platforms.
You will have to change trains en-route, probably at Agualva-Cacem Station, but you should check that at the Ticket office.
As regards prices of taxis, see this previous forum post:http://bit.ly/1em91td
As the above post notes, there are alternative routes to get to Sintra but the one I recommend above, imho, is the easiest.