To travel from Porto to Cascais by public transport one needs to route via Lisbon.
Between Lisbon and Cascais, the main (and practically only) means of public transport is the suburban train line "Linha da Cascais" which operates every 15 minutes or so between Lisbon Cais de Sodre Station and Cascais.
There are two main means of land-based public transport between Porto and Lisbon - Train and Long-distance coach.
Train: http://www.cp.pt
There are +- 19 trains per day between Porto Campanha Station and Lisbon Oriente Station. Most of these continue on to the terminus at Lisbon Stª. Apolonia , although there are two trains daily where one has to change to a local train at Lisbon Oriente to connect to Lisbon Stª. Apolonia.
To travel to Cascais, one has two options:
1) Disembark from the Porto to Lisbon train at Lisbon Gare Oriente Station:
Go down one level and find the CP ticket office who should be able to assist you with the purchase of a ticket that will allow you to use the Metro and suburban train from Lisbon Cais de Sodre Station to Cascais.
Descend into the bowels of the earth to Lisbon Gare Oriente Metro station.
Take the Red Line (Linha Vermelha) in the direction of São Sebastião.
Descend at Alameda Station and change to the Green Line (Linha Verde) in the direction of Cais de Sodre.
On arriving at Cais de Sodre exit the Metro Station Up two floors and you will be in the Suburban Station.
Take the train to Cascais.
2) Disembark from the Porto to Lisbon train at Lisbon Stª. Apolonia Station:
Exit the Station and take the Nº. 728 bus (every 15 minutes or so) in the direction of Restelo.
Descend at Cais de Sodre and follow the previous instructions.
Download Timetable here: http://bit.ly/14mAoOP
Bus:
There are two main bus companies operating long-distance bus services between Porto and Lisbon.
1) renex.pt
Renex buses depart from Porto Campo dos Martires and arrive at Lisbon Gare Oriente Bus Station (next to the rail station of the same name.)
Walk over to the railway station.
Descend into the bowels of the earth to Lisbon Gare Oriente Metro station.
Take the Red Line (Linha Vermelha) in the direction of São Sebastião.
Descend at Alameda Station and change to the Green Line (Linha Verde) in the direction of Cais de Sodre.
On arriving at Cais de Sodre exit the Metro Station Up two floors and you will be in the Suburban Station.
Take the train to Cascais.
2) www.rede-expressos.pt/
rede-expressos.pt operates +- 19 daily services between Porto GARAGEM ATLANTICO, RUA ALEXANDRE HERCULANO Nº366, 4000 PORTO and Lisbon Sete Rios Bus Station.
On exiting Lisbon Sete Rios Bus Station cross over to the rail station and descend into Metro. Take the Blue line (Linha Azul) in the direction of Stª. Aplonia and descend at Baixa Chiado. Cross to the Green Line (Linha Verde) and then continue to Cais de Sodre and follow the previous instructions above.
TAP - Air Portugal also operates several daily flights between Porto and Lisbon.
