When I go to my contacts, my FaceTime option doesn’t show up

FaceTime Audio is not available is some countries.

1. You can use FaceTime on any of these devices on Wi-Fi: iPhone 4 or later, iPad Pro (all models), iPad 2 or later, iPad mini (all models), and iPod touch 4th generation or later (only iPod touch 5th generation or later support FaceTime audio calling).
2. With a cellular data plan, you can also use FaceTime without Wi-Fi on any of these devices: iPhone 4s or later, iPad Pro (all models), and iPad (3rd generation or later).
3. When you tap a person's contact information, FaceTime tries to connect using any FaceTime-enabled numbers or addresses that you have stored for that person. To avoid placing calls to unintended recipients, make sure that your contacts are accurate and don't contain old or unused information.

Also know more

• Learn more about FaceTime activation alerts.
• Learn what to do if you can't make or receive a FaceTime call. 
• Find a missing FaceTime app. 
• Find out how to block FaceTime calls.

FaceTime isn’t available or might not appear on devices purchased or used in Saudi Arabia, United Arab Emirates, and Pakistan. However, with iOS 11.3 and later, FaceTime is available on iPhone, iPad, and iPod touch in Saudi Arabia. And with iOS 12.4 or later, FaceTime is available on iPhone, iPad, and iPod touch in Pakistan. Update your device to get FaceTime in Saudi Arabia and Pakistan.

Watch the GIF below. Press and Hold, Call or Video icon.