3 weeks off in Portugal, and have not booked yet...

1). Prices of flights to get there will be expensive at this late stage. Accommodation will be cheaper than in August, but choice maybe limited as lots of pensioners and golfers head out there for some late summer sun as soon as the children go back to school. Late September in some of the smaller resorts will have an end of season feel. The weather should still be OK but there is a greater chance of the occasional shower than in August.

2). Can't comment. I always fly direct.

3). The Algarve is all about the beaches and golf. If you want more culture, walks and history, you would be better in say Lisbon to the north. There is a limited bus service along the Algarve, but car hire would be best if you want to explore more. It takes c 2.5 hours to drive from one end of the Algarve to the other. Many of the resorts on the Algarve are purpose-built and lack the character that the towns of the north enjoy.

If it has to be the Algarve, I would suggest splitting your time between Lagos and Tavira.

Mr . V , I think your main issue with prices might be flights. We flew British Airways, but it looks pricy for September. Air Transat might get you cheaper to somewhere like Amsterdam or London. They don't fly from Calgary to Portugal unfortunately. There are no direct flights that I am aware of from Calgary to Portugal.
Otherwise, I would give it a try. Try looking for some of the smaller towns for more availability and better prices. We found booking an apartment in Lisbon a better deal, I have also heard the hostels are excellent there. Food prices are very reasonable in Portugal, so that you will be able to do comfortably on a budget.

"How to get there? What route did people go? London, Amsterdam, direct?"

Got to Kayak: http://www.kayak.com/flights. Choose "round trip." Put in "Calgary, Canada (YYC)" as your origin and "Faro, Portugal (FAO)" as your destination. Faro is actually in the Algarve, in the south; if you want to see other parts of Portugal, look into flights to Lisbon (LIS) in the center of the country, or Porto (OPO) in the north. If you want to start in one and end in the other, choose "multi city" instead of "round trip" (for instance, YYC to OPO as the first leg, then FAO to YYC as the second leg).

This will show you all kinds of available routings, along with the prices. You will have to change planes, possibly several times. Once you have an idea of what is available, you can sort by "shortest flight" or by "lowest fare" or other parameters. Furthermore, on the left of your screen, you can choose how long a layover you will tolerate, what arrival or departure times you want, etc.

Have you looked into AirTransat to London and EasyJet to Faro?

Thanks for all the helpful responses!
Still planning and coordinating.