1. Start with a clean slate. I always use a blank template and I always choose the default Ski theme. I think it's the cleanest theme and will give you a nice, blank canvas to build your site.
2. Ditch the sidebar navigation. I think horizontal navigation is a more modern look for your website. The downside to this is that you will have to manually add newly created pages to your horizontal navigation. (Sidebar navigation automatically creates that navigation tree for you.)
3. Create a custom header. I use a Chrome Extension called Page Ruler to get dimensions of my header area and then I use Google Drawing to create my graphic. I re-size my Google Drawing canvas based on what I measured using Page Ruler (sometimes have to reduce width) and then layer elements until it's just right. Then I download the graphic as a .png and insert it as my site header.
4. Clean it up. There are a few tweaks you can make to really clean up your site and make it look polished. The first thing I do is turn off the site name at the top of the page. Then, I remove the search function which is currently showing up over my header. Last, I remove the background color on the whole page. All of this is done in "Manage Site."
5. Configure the horizontal navigation. In step 2 I shared that you have to manually add new pages to horizontal navigation. While this is a small inconvenience, I think it's worth it to have a modern-looking site navigation system. You can add existing new pages to your horizontal navigation under "Site Layout." I am also a fan of centering the horizontal navigation instead of leaving it left justified. This can be done under "Manage Site" in the Themes, Colors and Fonts tab.
BONUS TIP! Adjust your page settings to create a cleaner, simpler page layout. I like to remove the page name from the top of the page and also disable commenting and attachments.
If you need additional help or have additional questions, feel free to contact me! Coming soon ... YouTube playlist with the above Google Sites tips! :)