Lisbon - cost to fly / favorite website to book fares

Unfortunately you don't say where you're flying from. I'll assume you're in the U.S.
Lisbon is not well served by non-stops from the U.S.
American flies seasonally from Philadelphia--whatever seasonally means.
United flies from Newark and seasonally from Washington Dulles.
Since there's little competition, the lowest airfares from the U.S. are not expected to Lisbon without going through other European Gateway airports.
MatrixITASoftware.com would be the best place for you to research airfares at legacy air carriers. Then go to their individual websites to purchase the tickets.
Expect a few European budget airlines flying a number of airports in the Northeast, the West and Florida much less expensively than American or United. You might have to go through Gatwick or Dublin or Oslo to connect with another budget European airline flight to Lisbon, however.
It all depends on what airports you can fly out of .

You might also check with TAP Portugal, national airline of Portugal. Direct flights from Miami, Boston, New York and Toronto. Multi stop from other cities.

Possibly, check Air Canada out of Toronto. Norwegian Air may get you to Oslo for cheap. United Air recently ran sales! I have not found a cheap flight that meets all my criteria in a long time!

Alas, flights cost what they cost; there is simply no rhyme or reason to airfares any more. Start by looking on Kayak, Skyscanner, and Google Flights. Also look directly on the airline websites, once you know who flies there from your destination.

If you have any flexibility on which airport you start from, be sure to look at all of them. If a flight to Porto or Lagos Faro would work for you, look at prices for these as well; again, for no clear reason, these may be cheaper. Of course, you have to add the time and money it would take you to get to Lisbon.

Sometimes you can save money with a roundabout routing or multiple connections; only you can decide how much hassle you will put up with to save money.

As said above, nonstop flights from North America to Lisbon are limited. If you have to connect and you have a choice, most people prefer to connect in Europe rather than North America. If you miss your connection and are booked on one ticket, the airline will get you to Lisbon on a later flight, at no extra charge. However, if there's only one flight a day from, say, Newark to Lisbon, then you have to wait a whole day. If you are flying through Frankfurt and the arriving flight is late, there will be several Frankfurt to Lisbon flights later that day.

EDIT to reflect the correct airport in the south of Portugal.

Your profile doesn't say where you are located, however, if it's convenient for you to fly with a connection in Toronto (Canada), there are overnight non-stop flights from Toronto to Lisbon on three carriers: TAP Portugal, Air Canada Rouge (not the regular Air Canada planes), and Air Transat (a holiday travel airline). Have a look at Skyscanner.ca for sample airfares and book directly with whichever airline you choose.

I second Phillip's recommendation for TAP, the national airline of Portugal. I have flown them twice and would fly them again in a minute if heading to Portugal.