Having worked in strategic planning for a Major and an Express carrier, airports try a variety of ways to get an airline's attention. Marketing support is pretty standard, except Vegas, they market the brand of the city which an airline takes advantage of for new service. They also will waive landing fees and terminal rents to lower start-up costs and hire consulting firms to run route numbers (Passenger and average fares, connecting opportunities, etc.) and economic analysis of a market's overall economy to pitch new service ideas.
The gifts they send also can be pertinent or comical. A three foot penny promoting an airport in Illinois as the Land of Lincoln went up on the wall as the worst we ever got. The coolest one, Akron, OH got us a ride on the Goodyear blimp. They also invite planning teams to personally view the city and its economic growth (cranes and new development are a plus.) The biggest pitch is to renovate or build new terminal facilities, but Southwest tends to be the only one who gets that royal treatment.
Talks between an airline and cities can go on for years. International routes tend to be the most prized. In the end, the odds of a market being profitable, how it helps the overall strategic picture (feeds other routes,) will there be a competitive response, and its future economic prospects are the primary driver of new service. Subsidies and marketing support are helpful, but won't overcome otherwise weak prospects for a market.
Alaska almost always starts a new market from Seattle, their home fortress. Virgin America focuses on LAX and SFO as JFK is slot controlled for adding new flights.