By plane
Spokane International Airport (http://www.spokaneairports.net) IATA code GEG for Geiger Field is about seven miles to the west of Downtown. It operates flights regularly to Seattle, Portland, Denver, Salt Lake City, San Francisco, Oakland, Sacramento, Minneapolis, Chicago, Las Vegas, Phoenix, and Boise. There are no non-stop international flights at this time though there were in the past to Canada. Flights to/from Canada normally go through Seattle most common, Minneapolis, or Chicago. If everything is full, you might have to go further south. Currently, there's no way to get to Calgary and Edmonton without flying to a quite out-of-the-way hub city. Although American Airlines has never had service to Spokane, it now code-shares with Alaska/Horizon Air.
By train
Spokane is located on Amtrak's (http://amtrak.com) Empire Builder route, which operates between Seattle/Portland and Chicago. Spokane is located at the point where westbound trains are split in half, one half going to Seattle through the Cascades and the other half to Portland OR-Vancouver WA along the Columbia River Gorge, and where eastbound trains from those cities are joined together to continue eastward. The Empire Builder connects with Amtrak's Coast Starlight in Seattle and Portland and with many other trains in Chicago. The Amtrak intermodal station, which is shared with Greyhound buses, is a new facility located downtown, close to major hotels and attractions.
By bus
Same goes for Greyhound. You can get to Spokane on Greyhound from just about anywhere in the U.S. Its station is the intermodal one previously mentioned.