Algarve and Andalucia - 10 days in July

Picking up a hire car in Portugal and dropping it off in Spain will incur a hefy surcharge because the car will have Portuguese plates and will have to be repatriated to Portugal. In any case, driving into Seville is for the madly suicidal. It will probably mean the end of a beautiful friendship.

There are buses from Lagos to Faro, and from Faro to Seville, and plenty of public transport options from there to Granada and Cordoba.

Lagos has a lovely beach. You could also go right up to Cape St Vincent, which is the most westerly point in Europe, and where Henry the Navigator had his School of Navigation.

Unless you're on an early morning flight into Faro and a late evening flight out of Malaga, you've only got 9 days. Allow a short half-day to get from Faro to Lagos where you're going to spend a 'few days'. Then you've got the best part of a day's travel from Lagos to Seville.
There's a direct service: choose betwen these two.

Leave your accommodation at around 0645
Dep 0730
Arr 1400
Get checked in to your Seville accommodation at around 1500

Leave your accommodation at around 1145
Dep 1230
Arr 1745
Get checked in to your Seville accommodation at around 1845
There's a later bus but I doubt you'd be interested.

Travel options between the Algarve and Andalucia.
http://thomasallen.home.sapo.pt/travel/001.htm#bus

How many days were you thinking of spending in Lagos and how many in Seville?

I realise your flights are probably already booked and paid for but I don't really understand your thinking. 10 days is just about adequate for Seville, Cordoba and Granada never mind gadding about in Portugal.

If you insist on going to Lagos (not the most logical direction if you are then going to head for Andalucia) then I fear your are going to have to ditch one of your Spanish destinations.

I agree with Stevegerms. Hence my question at the end of #3.

Seville, Cordoba and Malada will be very warm and full of turists, so if you wanna spend a few days relaxing you should go to Lagos and to Sagres. Nice food and friendly people. You should go there.

Here's what I propose. Let me know if this seems feasible:

3rd July - arrive Faro, take a taxi to Lagos
4th July - Lagos
5th July - Lagos
6th July - Lagos/Seville
7th July - Seville
8th July - Seville
9th July - Seville/Cordoba
10 July - Cordoba
11th July - Cordoba/Granada
12th July Granada
13th July - Granada/fly home from Malaga (late evening)

I was also wondering if I should head to Tariva instead of Lagos - which is better. We are in our late twenties and like a bit of beach, a lot of wandering around, good food and active adventures. Also whereabouts is Sagres - is that east or west of Lagos? I don't know anything about Sagres.

As I'm sure you're aware, Tavira makes more sense than Lagos in terms of the logisitics of your trip since Tavira lies between Faro and Seville. That makes the journey faster and cheaper to Seville and might also save you a taxi fare. Here's the bus timetable from Faro to Tavira.
http://www.eva-bus.com/home_pesquisa2008.php?lang=UK&servico=0&origem=Faro&destino=TAVIRA&Procurar=Ok

Buses from Tavira to Seville:
http://www.eva-bus.com/home_pesquisa2008.php?lang=UK&servico=0&origem=TAVIRA&destino=Sevilha&Procurar=Ok
*Partida = departs
*Chegada = arrives

Sagres is about as far west as you can get on the Algarve. It's 40 kms to the west of Lagos.

From what you've said, I think Tavira will suit you fine.

According to Google its about 90 km from Faro to Lagos - you sure you want to get a taxi?