You can change the exposure settings on even the most basic compact camera by using the exposure compensation feature. Exposure values are usually incremented in 1/3EV steps, so you may find, for example, that your first reading is 1/400th of a second at f/8, but that the second reading is 1/320th of a second at f/8, which is an increase in exposure value of 1/3rd of a stop, or 1/3EV. Likewise a change in shutter speed from 1/400th to 1/640th of a second is a reduction in exposure of 2/3EV. Exposure compensation is also incremented in 1/3EV steps, so in the first case setting it to -0.3 will change the exposure to the same value as the first reading, while in the second case a setting of +0.6 will do the same. For a more detailed look at exposure values, see my previous tutorial on the subject here.
Having made these preparations, you're ready to take your shots. Many cameras with panorama stitching functions also allow you to pick the direction in which you wish to pan, although the default is usually left-to-right, so pick your starting position and start shooting. After you've taken the first shot you'll see the right-hand edge of it super-imposed as a semi-transparent overlay on the left-hand side of the monitor. Simply rotate your camera on the tripod until the details on the left of the scene match up as closely as possible with the overlay. If you're using an automatic compact, check and adjust the exposure value, and then take your second shot.
The more shots you take, the longer it will take your editing program to stitch them all together, and the more likely it is that the results will be unsatisfactory. Although it is possible to stitch together dozens of shots, unless you have a very powerful PC with a lot of memory it's a good idea to limit yourself to a maximum of five or six shots. At a focal length equivalent to 50mm your camera has a horizontal angle of view of approximately 47 degrees, so even allowing for the overlap between shots, six frames should be more than enough to cover 180 degrees.