The Sensor is in exactly the same place that the film would be, if it were a film camera.
As mentioned, it's behind the mirror and the shutter (when looking into the front of the camera).
Yes, DSLR sensors will need to be cleaned from time to time...and if you aren't 'all thumbs', it's not too difficult to do yourself.
To see how much dust you have, put your camera into aperture priority mode and set an aperture of F22 (give or take). Then take a photo of a smooth uniform surface like a wall. Dust on the sensor will show up as grey specks or blobs.
Search the forum (or Google) for 'sensor cleaning' to find more info.