When the card is inserted in the camera for the first time, are you able to format the card using the Format option in the Options Menu? If this isn't working either, I would advise trying another brand of SD card (perhaps in a local photography store to save the extra hassle of trying to get hold of various media) and if it literally will not read any type of SD card whatsoever, this may indicate an internal fault within the camera.
If the latter is true, Sony Customer Services can be contacted using the details in this forum thread:
On my Sony Nex-5 memory card i cannot see my videos when connected to a laptop while the still downloads fine. They are there when the card is in camera but shows no movies when connected to computer. What could be the reason? The card is genuine as i had it with the camera from a known retailer.
Stills are often visible in your computer's file browser whichever way you download them, but your NEX-5 videos will be in AVCHD format, which creates a number of files not immediately readable by the computer. I need to ask you some questions before I can help:
Which image browser are you using?
Are you using a Mac?
If PC, which OS is it running (Vista, XP, Windows 7, 64bit or 32bit?)