The "white plug" is a fairly modern DVI connection. The "blue plug" is an older VGA connection. DVI supports much higher resolutions, so you'll probably need to continue using it for the 27in monitor. The two dual-display modes are "mirrored," where the same image is displayed on both screens, and "extended," where the computer treats the monitors, regardless of their sizes or resolutions, like one big one. Extended is the mode you want, since it will let you display different things on both monitors.
I can provide instructions for setting up an extended display, but only if you use a PC. I remain stubbornly in the Windows camp.