We are looking for a private driver to take us from. Lagos to Lisbon and that might stop along the way for tourist sites. I know that the website Daytrip exists but I was wondering if there are other people who do this sort of service to compare price and stops. Please offer a recommendation if you have one. Thanks.

The cost for a cab for a distance like this, of over 300K is somewhere between 300 and 500 euros. If you're happy at that level rather than taking the train at around 20 euros per person, then I'd ask at your accommodation for a recommendation for a local firm.

also, I have a picture of the Brooklyn bridge you can buy ;-)

Daytrip.com is a consolidator (like Get My Guide) that contracts with private transfer companies/drivers, so you will pay more using daytrip.com than dealing directly with a private, local transfer company. I've never used a service from Lagos to Lisbon, but Rubicund's estimate seems very correct.

Most Algarve transfer companies are based in Faro (not Lagos), providing services primarily from the airport.
As Rubicund suggests, I would ask your hotel (or villa owners, or B&B hosts) for a reliable local recommendation, a transfer service that will go beyond the Algarve and not just to from the Faro airport or to/from your lodging to a golf resort.