I own a Nikon Coolpix L120 camera. It's fully automatic. I am really interested in photography. Can anyone suggest me some exercises or experiments to do with this camera so that I can make a good use of it for learning?
My first recommendation is that you familiarize with the basic photography concepts, starting with the exposure triangle. All cameras are based on these principles, so it is important to understand them, even for the cameras like yours that don't let you set some of these settings.
Understanding the above will make you realize that calling your camera "fully automatic" is an exaggeration. While it is true that there is nothing even remotely close to manual mode in this camera, there are many ways in which you can affect how the camera takes pictures, so you do have some amount of creative control. Examples:
As far as inspiration to learn, my recommendation is that you do a 365 project. This is simply the task of taking one picture per day for an entire year. The pictures can be thematic (i.e. self-portraits, landscapes, etc.) or just whatever you can think of each day. You can upload the pictures to Flickr and share them on groups devoted to 365 projects (there are many of them), or you can just do it for yourself and keep the collection of pictures privately. But the fact that you have to get out there and take a picture every day will give you the motivation to improve and learn.
I also have to give you the bad news. If you follow my advice and get into the habit of taking pictures, you will eventually realize that you need a better camera. So you may want to start saving, for when that moment comes ;-)
Shoot pictures, and then shoot pictures, and after a while try to read, or re-read, the manual for your camera cover to cover.
If the reading is too terse, buy a book that explains how to get the most out of your specific camera model. ("Magic lantern guides" for instance ... I haven't read any of these books... yet. But that's not the point).
THe point is that I, after seriously studying the manual for my camera (Olympus pen E-P1), I discovered many hidden features and even basic concepts of operating it. DIgital cameras are quite complex beasts if you go beyond the basic shooting mode.
A simple experiment with a technical focus: You could try to test how fast your memory cards really are. http://www.smallcamerabigpicture.com/olympus-omd-em5-memory-card-test/
You can search for tips for using compact camera on the internet, but IMO the best way to understand what your camera is capable of and to find personalized tutorials, is to look for images taken by your camera and maybe ask their photographers about their techniques, for a start check Nikon Coolpix L120 on Flickr.
There is already an answer covering technical aspects, however, I want to (at least try) to focus on the other side of photography: The artistic one.
Composition. My sugestion on the excercise is to take several pictures of the same subject, placing it in different points of the frame. (I recommend to forget about the pictures for a while, a couple of days would do) And review them later, decide where the particular subject fits better, where it draws your attention or where it looks dull. Decide whether you follow the Rule of Thirds or not, of if you preffer The Golden Ratio or Golden Section. Also try switching from landscape to portrait framing, and vice versa. Do not follow any rule, take the both shots, make a safe copy, forget about them and review later! :)
Ligthing! Not all about a photo is about the amount of light on the picture (Exposure, over exposure, underexposure) but also about light falloff, light, shadow and gradation is extremely important, to the point that most of the time it defines the feel of a photo. The correct light setup can make a person hate a portrait of themselves or fall in love with it. Make several essays, ask a person to be your model and photograph them in several setups. By the side of a window, against a window, in sunlight, in the shade. If you use lamps (can be household lamps) take shots with the light coming from the front, from the side, from 45 degrees, from 135 degrees, from below, from behind, etc. Also try to shoot photos in a very dark room (if you have access to a black one, the better!) so the only light on your subject is coming from the lamps.
Flash management! You can modify your flash without tampering your camera. You can use difusers made from household materials. You can use trasnlucent papers or plastic bites. You can use white materials or tinted ones. Play with your camera's white balance to enhance the effects. You can use a small handheld mirror to bounce the flash on the ceiling or on a wall. The color of the ceiling, wall or whatever object will affect the photo, for good or for bad. Try making cardboard cutouts to create weird shadows with your flash or to control where your flash iluminates the scene and where it doesn't (Create a small snoot). Just a tip: Whwnever you put something in front of your flash, don't put it right against the flas lens, you may burn something, and it could be the flash (some of them generate a small heat wave). Instead keep a distance of two centimeters or more, but to avoid flare, whatever you place in front of the flash, place it no further than the lens front element.
Production! The fact that you are not using a DSLR must not prevent you from arranging your Photo Studio! Plan your sesions, don't just press the shutter. Take into account location (the best room in your house, for example), the furniture, accesories, clothing, makeup, time of day (specially if outdoors). If taking portrait shots, consider the animic state of your model. Arrange all your setup prior to bringing the model(s) in, so you don't exhaust their patience, make them feel relaxed so you get better smiles. Know your subject so you can predict reactions and catch the perfect moment where the smile is natural, etc...
Gather knowledge Of course you can read books, but you can also apreciate photograps in magazines, or expositions. Look carefully for "photographic composition" in movies (Have you noticed there is always a "Director of Photography" among the credits?). Go to museums and apreciate not only photography, but also paintings, sculpture. Analyze them. Do you like them? Why or why not? How are the elements arranged in the composition? Perspective? Space management?
Chances are you can replicate much of these characteristics, effects and techniques, because they all are external to the camera, but if they are in the image, they make part of the photo, and can make it great or dull.
All of these excercises, combined with the more technical ones will lead you to really know your cameras strong points and limitations, you will get to know it characteristics beyond the specifications in the manual so you can better exploit your valuable resource.