I did a bit of googling hoping to find a post on IDEA shortcuts similar to Jeff's post on Visual Studio shortcuts (Visual Studio .NET 2003 and 2005 Keyboard Shortcuts), but didn't really spot anything that helped. Hopefully the answers to this question will fill the void.
This is probably dependent on your development environment, but when doing BlackBerry development I code in IntelliJ and still compile and debug through the BlackBerry JDE. Ctrl + Shift + C is vital in this case as it allows you to copy the full path of the file you are currently editing, making it easy to open the file in the separate dev environment.
Yes, Ctrl + Shift + A is the most useful one. It's a meta shortcut
Within the tool: "Help -> Default Keymap Reference" (as suggested by krosenvold)
On the Internet: "IntelliJ IDEA Developers Documentation"
Windows, Linux and Mac keyboard shortcut reference cards for versions 8, 7, 6, 5, and 4.5.
(I knew I had seen and used these reference "cards" before! ;-)
One of my favorites (not shown on the JetBrains cards) is "Ctrl+Shift+{Up or Down Arrow}" to move lines/methods up and down in the source code.
See also "Intellij Idea Keyboard Shortcuts" page on Ward's Wiki.
Shift+Delete deletes the entire line (will 'cut' it to clipboard)
Ctrl+Alt+L to reformat and optimize imports
Ctrl+Shift+J to join lines (pull content of next line up to current line).
F7 F8 F9 for debugging
Ctrl + N : Open class.
Alt + F7 : Find usages.