I used to upgrade my point-and-shoot camera frequently. I thought that the nifty features of newer cameras would improve my photos. Sure, equipment (camera, lens, accessories) can make a big difference in results. However, equipment without technique will not improve shots, and as a beginner, the limiting factor in the quality of my pictures was my technique (or lack thereof). To illustrate the difference that technique makes, take a look at the shots below:
The shot at the top is with a Nikon D300, a Tamron 17-50 2.8 VC lens, and a Nikon SB-800 flash. The camera, lens and flash are all prosumer quality and have a total cost of easily more than $2500. The shot at the bottom is with an average 3-year old point-and-shoot camera (the Casio Exilim EX-V7) and the SB-800 flash. The difference between the two shots is purely due to technique (specifically, lighting).
In summary, if you're new to photography and want to improve the quality of your pictures, the biggest improvement will come from improving your technique first. If you want to know where to start learning techniques, check out these resources:
Roadmap for Family Photographers
Beginner's Simplified Guide to Exposure
Introduction to Lighting for Family Photographers