Web safe palletes are not manadatory, they just help avoid "funky" outputs by combining safe colors in an attempt to acheive your on screen (non-web safe) color.
I am not familiar with Corel Elements, so can't comment specifically on that part of the question. Corel Draw supports a web export, which does precisely what you are asking, but in most cases, I find it is better to start a web page from scratch then trying to manipulate the Corel output. I have had a few instances where I had to convert a print catalog to a web page catalog, and had no choice but to use the Corel function. I found it better to use a series of small PDF's then to try and use Corel's code.
If there is a similar function in Elements, I am not sure it would be a good way to learn web design. IMO if you want a WYSIWYG (What You See Is What You Get) editor, search specifically for such a tool, and use that. I personally like the Macromedia products for that, but they are advanced tools. I am sure there are some good freebies and/or cheapies that would suffice to get your feet wet. You might check some of the web design forums. I would also avoid using MS Word, even though it does a suitable job converting the file to HTML, it is sort of a hybrid XML output, and very hard to edit. Many web servers choke on the MS code, and some sites even offer tools to remove the extraneous code from MS outputs.
My advice is stick to Corel for image enhancement, button creation, image conversion, resizing, etc. To truly learn HTML, start with Notepad. To get a web page of the kids and the dog online for Grandma and Aunt Bernie - go with a WYSIWYG with an easy to use interface.
