Getting activation unsuccessful when trying to activate iMessage or FaceTime

First, check to know whether you have an SMS plan enabled at least for One SMS with your carrier. Contact your Carrier (Cellular Service Provider) the agency who issued the SIM to you

If your iPhone is used with a carrier not found in the wireless carrier articles, make sure you can send and receive international SMS. (One SMS Only)

See the pic below

4 Step Process --> Turn off and restart iMessage and FaceTime (The order/sequence is very important)

1. Go to Settings > Messages and turn off iMessage.
2. Go to Settings > FaceTime and turn off FaceTime.
3. Restart your device.
4. Turn iMessage and FaceTime back on (Watch the GIF hereunder)