Apple ID works on everything except App Store

Some 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   *

Restart your iPad.   *

Try a forced restart. See the section towards the end of the article.

Reset Device. -

If necessary, contact iTunes Support for assistance. See the Apple Support articles below for contact information.   

iTunes Support – world wide   -

Apple Support schedule phone call     -

Billings and Subscriptions   -

If necessary, use Twitter or Messages.

Apple on Twitter

Messages chat with Apple