Is there a shortcut key to move a task to a new Project?
Unfortunately, there's currently no keyboard shortcut for moving tasks to a different project. You can refer to this link for all available keyboard shortcuts that you can use: https://todoist.com/Help/KeyboardShortcuts.
Please note though that if you have one or more tasks selected, you should have the "Move to Project" option available at the top of the screen. This should allow you to quickly move tasks around. Alternatively, you can also drag and drop tasks to different projects. I know it's not the same as a keyboard shortcut but I hope those two options help.
I hope this feature is added in a near future.