Watch the preliminaries.
Americans and Canadians are not allowed to purchase tickets via the official Sochi 2014 sales website. Instead, tickets must be purchased from CoSport, the authorized Olympic ticket reseller for North America. Because of this arrangement, you will pay considerably more for tickets than Russian citizens. But CoSport's priority is to sell entire travel packages to the Games. Even if you don't need accommodations, you can still purchase individual event tickets from its website. Just be prepared for some sticker shock. Tickets to the medal rounds of major events can go for as high as $800.
As of now, there is no shortage of less-expensive tickets for preliminary events. Don't get discouraged if the sport you want to watch is too expensive. After you have your tickets, you'll need to apply online for a Spectator Pass. This photo ID is required to gain entry to any of the venues, making the event hell for scalpers. There is no charge for this pass.