I just got OS X Mavericks and the new notification bar (as seen in the right of this link) in the top right is bugging me. Is there a way to remove or move it, so the search bar is the first thing in the top right?
You can do this with Bartender.
This allows you to rearrange and hide menu bar items, including items that Apple prohibit from moving such as Spotlight or Notification Center. It also lets you move items to a secondary bar or to be hidden until a keyboard shortcut is pressed.
To kill the notification bar, type this:
launchctl unload -w /System/Library/LaunchAgents/com.apple.notificationcenterui.plist && killall NotificationCenter
If you want to renable it, type:
launchctl load -w /System/Library/LaunchAgents/com.apple.notificationcenterui.plist && killall NotificationCenter
As for moving it, so that the Spotlight function is first, I do not know. You can easily access Spotlight by keyboard using ⌘ Command+Space and that will quickly open Spotlight, and you can just start typing without having to move your mouse.