If you're getting serious, but you're not yet sure just how serious you're going to get, the answer is yes, you can get by with Elements. For now.
Photoshop's "junior" version, Elements 5.0, is a great program. And it has a lot of the most commonly used Photoshop tools: the awesome Healing Brush and its cousin the Spot Healing Brush, the Magnetic Lasso, and the Clone Stamp.
Many of Photoshop's picture-fixing functions are present, as well, such as Levels and an abbreviated version of the wondertool, Shadow/Highlight. Plus, it has a few features that ought to be in Photoshop itself, like Adjust Color for Skin Tone and a new Black-and-White Converter. You can even edit with Layers and create Adjustment Layers to make nondestructive changes to your pictures.
Elements is also full of How-To's and tutorials, and the Windows version comes with an excellent photo organizer and browser. Folks who are into sharing their pictures can make sophisticated Flash slideshows and share them right from the program, and make and save scrapbook-like layouts to a new multipage format.
What you won't find in Elements are tools you'll need in order to do sophisticated photo restoration or retouching: Masks, a compositing essential, are present only with Adjustment Layers; the Patch tool is conspicuously absent; and, while Curves is newly available in Elements, it's in a truncated introductory form. You'll need Photoshop for lots of other tools that pros use, such as CMYK conversion, the Pen tool, and the ability to work on individual color channels.
You should start with Elements 5.0. It's reasonably priced, and if you're a newbie it won't overwhelm you. Mastering it will make you familiar with the Photoshop MO, so when you're ready to step up, it will be easy to incorporate all of the things the complete Photoshop offers. By that time, a new version of Photoshop will probably have been released. This way, you'll save yourself from having to upgrade just as you're beginning to figure out how the program works.