After you submit your website to the search engines (which should be free), there are several things that you can do to increase your rankings and your likelihood of coming up in searches.
1. Update your website regularly by adding new content and changing content so that when the search engine crawlers visit your website, they will log the changes. Put "keywords" in your text. Basically, this means to include words or phrases that you think people would use to search for a website like yours in the body text of the pages of your website. Obviously, that means saying Egyptian Mau a lot, which shouldn't be a problem for you. Don't overuse keywords, but don't forget to include them!
2. Add links to other reputable websites, especially websites with high Google PageRank. If you have a Google toolbar on your browser, it should show you the PageRank of any website you are on.
3. Get other websites to link to you, like other breeders, and link yourself on forums (put your link in your forum signature, for instance) and social networking sites like Myspace or Facebook. For instance, every time you put up some new photos, put another link on Facebook saying "I just put up new pictures on my website, check them out!" More visitors and more links mean higher PageRank for you!
Keep in mind that it is going to take TIME above all for your website to move up in the rankings. The age of a website also adds to its rankings on search engines. If you have any other questions, feel free to PM me. Maybe I can help. I did a lot of the search engine optimization for the websites at my last job, so I have some experience in this. At the very least, I can probably find you some articles that you can read about it!
As far as the website goes, I dunno what a breeder's website ought to have. The other breeders here can probably help you better than I can with that.
The only thing that I can think of is that while the template is pretty, you ought to have your kitties instead of butterflies at the top.