Yes, you can just rotate the camera to take pictures. Try to keep the camera in the same place. When I shot with a P&S, I tended to put the tripod hole on my left thumb, and spun it around that. Keep an eye on whether you're staying level with the horizon. You don't really need a tripod if there are no nearby objects you want in the panorama.
Shooting in Manual mode to lock the exposure, manual focus to lock the focus point, and with a non-auto white balance to lock the color temperature are also good ideas.
You want to overlap the frames by about a third. The overlap eliminates the possibility of missing coverage, and can help you erase moving objects/people in the shots--if they're in the overlap areas, if one shot has them and the other shot doesn't, you can use masks/layers to choose one or the other to eliminate ghosts and clones.
It's a good idea to cover more of the image you want than less--chances are good after you're stitching that you may have to rotate to straighten the horizon, or that the distortion needed to make the photos fit will require you to crop the pano. Having more coverage gives you more image to work with. Multiple rows is not a bad idea, or shooting with the camera in portrait mode to get more vertical coverage.
For stitching, depending on what kind of pano you're shooting and how much control over it you want, you can use a free push-button solution like
Autostitch (or if you have a Canon camera, Photostitch which came on the CD that came with the camera), or a more sophisticated stitcher, like
Hugin.
The main steps of stitching involve loading the images into the stitcher, arranging them, (in the case of Hugin, picking control points) and then stitching. From there, you can open the panorama in an editor, like Picasa or the Gimp, and adjust color/contrast, etc.