Without a doubt the most enjoyable way will be by hire car.
There is no convenient train services (you would have to go via Madrid and Seville), nor flights, (you would have to go via Madrid.)
As regards buses, alsa.es operate Lisbon Oriente (10 minutes from the Airport by cab) to Seville Daily via the Algarve.
LISBOA OR. SEVILLA 09:30 17:45 Euro 76,00 Daily Return
LISBOA OR. SEVILLA 20:45 05:45 Euro 76,00 Daily Return
TG Comes (http://www.tgcomes.es/) operate coaches between Sevilla to Cadiz every couple of hours.
On weekdays there are services leaving at 19:00 and 07:00 which would tie in with the arrival from Lisbon.
Return Price would be about Euro 23:00. Transit is under 2 hours.
Your round cost by bus would therefore be about Euro 100.00. For two people this would work out at Euro 200.00
The problem with this route is that it involves around 12 hours travel each way in a bus.
By Car, you have the chance to be more flexible and see more of both Spain and Portugal.
A Scenic route which I have driven much off as as follows: (Courtesy - Google Maps/Tele Atlas)
Driving directions to Cádiz, Spain from Lisbon Airport
521 km – about 6 hours 31 mins
See: Google Map at:
http://maps.google.com/maps?f=d&saddr=Aeroporto+de+Lisboa,+Lisbon,+Portugal&daddr=IP2+to:37.874853,-6.514893+to:Cadiz,+Spain&hl=en&geocode=%3BFRSgRAIdgNWH_w%3B%3B&mra=dpe&mrcr=0&mrsp=2&sz=7&via=1,2&sll=37.605528,-7.602539&sspn=4.046645,9.437256&ie=UTF8&t=h&z=7
This routes you via the Alentejo region of Portugal, The town of Beja, the town of Serpa, (Great place for layover - see Casa da Serpa in the right hand panel), Seville in Spain, and Cadiz.
As regards Car Hire, I would suggest that you look at Auto Jardim - website here:
http://www.autocarhire.net/
They have offices at Lisbon Airport. Make sure that you tell them you are travelling to and from Spain.
Also remember we in Europe mainly use manual shift vehicles. Make sure if you need an automatic to specific this.
In November this year, rental prices for a week without optional insurance etc start from Euro 100 for a small hatchback.
