You tap the blue + plus sign in order to open a new tab. It just so happens that your favorites will always show when you open a new tab. You just start tying the web address in the URL field to get to the website of your choice.
You cannot adjust the colors of the icons and make them completely transparent. The colors of the icons or folders will be shaded with the colors of the wallpaper due to the translucent nature of the icons. The only fix for this is to change the wallpaper to something that makes the folders and app icons more pleasing to your eye.