We did a trip around the same time last year, 14 nights. We spent 5 nights in Lisbon, 2 in Sintra, 5 in Porto, 2 in Peneda Geres, and 3 in the Algarve. Although going north to Peneda Geres was very interesting, it also rained the entire time, so not great for sightseeing.
We flew between Porto and the Algarve (Faro) and drove back to Lisbon, which saved us a day and a half. To minimize backtracking, how about doing Lisbon, then Porto, take 3 nights to drive down to the Algarve, stopping each night at a different town along the way? (i.e., Coimbra, Castelo Branco, Evora [these are just examples plucked off the map. You may find more interesting towns]). After the Algarve, the drive back to Lisbon is about 3 hours from Lagos.
A lot of people on this forum dismiss the Algarve, but we liked it a lot, and we didn't spend much time on the beach. Maybe because after all the rain up north, it was nice to get somewhere warm and sunny! The Seven Hanging Trails is insanely popular, but justifiably so; the views are lovely. We stayed on the eastern end, and enjoyed the Ria Formosa. We only got to spend a couple of hours in Sagres (we were driving back to Lisbon) but would have liked to have spent more time. We liked Tavira, Loule and Silves (although I have to admit that by the end of our trip, the castles were all starting to look the same). Southern Portugal is where most of the agriculture is, so you could find markets with great produce.