How do you take a 15 or 30 minute exposure with the Nikon D5000 and how do you set the remote up to snap the picture in this kind of mode?

For really long exposures (longer than 30 seconds) you need to set the camera to the Manual exposure mode and set the shutter speed to bulb. This keeps the shutter open as long as you hold down the shutter release button. Since this is inconvenient, a much easier way is with the remote.

Press the [info] button to bring up the shooting information display. Press the [i] button to put the cursor into the display. Use the cursor keys to move to the release mode (fifth item from the top along the right edge) and press the OK button. The Quick-response remote mode (marked with an icon of the remote control) fires the shutter when you press the button on the remote. the Delayed remote mode (marked with the remote and "2s") gives you two seconds to hide the remote behind your back when taking self-portraits.

In either of the remote modes, when the shutter speed is set to bulb the shutter opens when you press the button on the remote and stays open until you press the button again (or the camera battery dies). This way you can press the button, go and do something else for a while, then come back and press the button again.

Information on the release modes is in the "Release Mode" section of the manual (page 65 in my copy). Information on using the remote control is in the "Self-Timer and Remote Modes" section of the manual (page 67 in my copy).