Suppose I am watching a video on YouTube and want to watch another video. Is there a way to go to the search bar directly with a keyboard shortcut? I don't want to grab my mouse, focus on the small bar and go back to the keyboard again.
This depends on your browser.
For Firefox, while a video is playing, press the TAB key to get the cursor in the search bar.
For IE9, while a video is playing, you need to press the TAB key 5 times to go inside the search bar.
If you are within the focus of Flash, all your keyboard shortcuts do not work. All of these are limited to Flash, so tab will switch between different features of the player. (See this Superuser question.)
If you do alt+tab and then alt+tab again to regain focus of the browser, your keyboard controls are back; maybe a single tab or couple of tabs after that will take you to YouTube search.