This depends on what you plan to do with the images...
By natural, I mean staying true to the level of quality the JPEG provides in respects to white balance, exposure, etc.
No, you will not yield significantly better images. You will, however, yield more experience in photography by obtaining a better understanding of the limitations of your camera's sensor. Your post production workflow will improve and you'll gain a better perspective of what degree of manipulation is possible. This means that you will become more experienced in post production, but it will not necessarily yield better photos.
Now, your question asked if you would yield "much" better images with RAW, but I should say that you will yield slightly better images. Sharpening, noise reduction, and basic exposure changes will yield slightly higher quality images when printing larger than say 8x10.
That being said, you have more artistic capability when you shoot RAW. RAW lets you really maximize the dynamic range of your sensor by pulling in shadows, manipulating specific colors (saturation, luminosity, etc), and more. If your intent is to lean more toward artistic images with drastic modifications, then RAW will provide much more flexibility and yield better photos.
You mention HDR, which I would consider to be an artistic edit. In this case, yes, you will yield significantly better results when shooting in RAW over JPEG. This is due to HDR software using all information in an image when creating the final HDR result. Depending on the level of HDR you are going for, you may realize that a single RAW will produce the same results as merging multiple JPEG files.