You'll want to start off with a Seadragon.Viewer. To control zoom and pan, access your viewer's Seadragon.Viewport. For general behavior configurations, see Seadragon.Config. To customize the display strings, use Seadragon.Strings. For simple scenarios, these classes are all you need.
On the more advanced side, Seadragon.Drawer does the graphical heavy lifting. You can use its addOverlay() method to add overlays to your image. To add custom controls, use Seadragon.Viewer.addControl(). Seadragon.Button and Seadragon.ButtonGroup can help out with that if you want. Write your own Seadragon.TileSource if you want to handle custom content types; Seadragon.DziTileSource handles Deep Zoom Images.
Seadragon.EventManager, Seadragon.ImageLoader, Seadragon.MouseTracker, and Seadragon.Utils are general-purpose utilites. Seadragon.DisplayRect, Seadragon.Point, Seadragon.Rect, and Seadragon.Spring are basic types. Seadragon.Debug and Seadragon.Profiler are just for debugging.
For more general information, see our library how-to's.
All methods and constructors which take an HTML element can also take a string with the element's id instead. This is done through the use of Seadragon.Utils.getElement().