I can’t enter my Apple ID password to sign into iMessage or FaceTime

See these documents:

How to set up and activate iMessages for iPhone and iPad https://www.imore.com/how-to-activate-imessage-iphone-ipad

If you can't sign in to your iMessage account or FaceTime on your Mac  https://support.apple.com/en-us/HT204408

Did you get an error when you tried to activate? See If you get an error when trying to activate iMessage or FaceTime https://support.apple.com/en-us/HT201422. 

Use these support options if you need more help.

Get a call from Apple support to help solve your problem using this link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362 Be patient. Sometimes phone support is not available.

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter https://twitter.com/AppleSupport There may be a wait of an hour or so.

Apple ID account security for country-specific and region-specific help

 https://support.apple.com/en-us/HT204169

Some other things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Backup - Restore your device from an iCloud or iTunes backup.