The usual misconception that entry level photo hobbyists have is that they need a better camera to take better photographs. This is not necessarily true. Even the oldest model of a point-and-shoot camera can take better photos if the photographer knows some basic digital photography techniques. At the same time, the owner of the most expensive digital SLR is wasting his or her time without knowing how to take pictures properly.
If you have just started with photography, upgrading your point-and-shoot camera right away is not required. You do not need to rush to a professional photography shop and immediately swipe your credit card for that high-end digital SLR. It is better to hone your skills with a point-and-shoot camera before buying a new one. Again, it is all about digital photography techniques and not the equipments.
Do not push your budget to the limit. Once you are ready for an upgrade, you can even buy cheaper cameras by buying previously owned cameras on mint condition. There are also gadget shops that allow you to exchange your point-and-shoot for an entry level digital SLR for an additional cost. Otherwise, if you do not have a camera yet, opt for a secondhand entry level digital SLR that usually costs almost the same as a brand new point-and-shoot camera.
It is not about the features of the camera anyway but how you are capable of maximizing these features to create acceptable photographs. Before actually learning digital photography techniques elsewhere, you have to get to know your camera by reading the instruction manuals with the gadget on hand. Try to take the same subjects with different modes and effects to find out how to use each button. The rest will follow once you have mastered handling your camera.