SIM Activations can take between 30 minutes and 24 hours to complete, when the activation is fully completed under your dashboard
here http://www.giffgaff.com/dashboard your new allocated number will be shown along with any airtime credit or active goodybag bundle. I suggest you turn your device off and on again periodically to see if that helps.
If the SIM activation is not completed after 24 hours then i suggest you contact a giffgaff agent to investigate. To contact a giffgaff agent please follow this https://giffgaff.com/support/ask/activationslink.
As agents deal with many queries, it can take up to 24 hours to get a reply, but usually you get a reply in a couple of hours.
Any replies from agents can be found on your "My Giffgaff" page or the page can be accessed directly following this https://giffgaff.com/support/questions link.
If the SIM is activated then check the signal in your area to check for any faults with the network in your area. You can do so by following this link https://giffgaff.com/coverage-and-service
PLEASE NOTE: the checker only gets updated once an hour so if your problem has just started, or you’re not sure how long it’s been happening, if it says all is good, keep trying for a while to see if it gets an update.
If the signal in your area is fine then you should try a manual roam which will update your connection with the network. Click on this link to view how to do a manual roam http://community.giffgaff.com/t5/Using-giffgaff/Force-the-change-of-SMS-settings/ta-p/4871794
and try turning your device OFF for a couple of minutes and then turn it back ON again to force the device to refresh the network signal.