I've got two suggestions...one for the quick-n-easy method, and one for the ever-so-slightly more advanced method.
First off, your camera is fine for night shots - very good in fact. There are two different kinds of night shots and people sometimes confuse the two: Night shots with slow shutters, and hand-held night shots. For the first, any P&S camera can produce a very nice, printable result that should make anyone happy and look like a pro took it. For the second, your camera won't be capable, and you'd need a DSLR with a good low light lens.
Fortunately, when most people think of night shots, they think of the first variety.
For the super-simple method, turn your camera dial on top to the 'Night Scene' mode - it's the one that looks like a silhouette of a person in a box with a star in it. This mode will automatically set the camera to slow shutter speeds to capture more ambient light. It will also fire the flash if you don't disable it - so either turn the flash off if you know how, or just let it fire and make sure noone is in front of you. The most important thing is: Do not move the camera at all! Lay it down on a flat surface, or a tripod if you have one, and use the self-timer to trigger the shutter (even your finger pressing the button can move the camera).
Now, if you're willing to just try to play with the settings a little bit...the best thing to do would be to manually set your ISO to the lowest possible setting (there's an ISO button on the camera...press it, then use the 4-way selector to choose '80'...that's the lowest ISO, which means less noise in the shot). You will need to at least be in 'P' mode on the dial on top of the camera (Auto mode doesn't let you change any settings, whereas P mode works exactly the same as Auto mode, but allows you to override some of the settings). Don't worry - the camera will still make all the decisions and set everything except that one thing you changed (ISO).
From here, you can actually just lay the camera on a surface, aim, use the self timer, and press the shutter - the camera will take the picture and automatically use a longer shutter speed to get enough light to compensate for the low ISO. That's when you get those lovely, smooth, bright night shots that look so good.
As you get more brave, you can start to experiment with Tv mode - that will allow you to manually choose a shutter speed of several seconds to get just the shots you want. It's not as hard as it sounds!