Thanks to the folks on this forum, I have re-tooled our trip. We fly into Madrid. Fly out of Lisbon 12 days later. Let me know if you think the following allocation of time makes sense:
Fly into Madrid, arrive a.m. Spend day 1 in Madrid.
Stay in Madrid for Days 2, 3 and 4. (Three nights total)
Overnight train to Lisbon for night 4.
Pick up rental car and head to the Algarve for days 5, 6, 7 (2 nights.) then back to Lisbon.
Return to Lisbon, drop off car and stay for 5 nights (days 8,9,10,11).
Leave very early on Day 12.
I am thinking a side trip from Madrid to Toledo. And a side trip from Lisbon to Sintra.
All comments and suggestions are welcome. thanks.