I got my unlocked iphone 5 today. I carefully cut down the SIM to fit and changed the settings to giffgaff's. However, I can't make or receive calls and text messages, access 3G etc. I can, though, call giffgaff voicemail - I guess that means the SIM is not damaged? The SIM is activated and topped up, too.
Try go to Settings -> General -> Reset -> Reset Network Settings.
This won't delete any of your data, however, you might have to reconnect you wifi and input APN settings again.
If that still doesn't work then I suspect you might have damanged your SIM.
Should you be experiencing poor signal strength or no service, it is occasionally necessary to perform a manual roam on your device to de-register it from your network provider and then re-register it again.