How to install apps in App Store with a new Apple ID that was already signed in in my new iPad?
That is normal. Your Apple ID is different from the ID you use to sign in to your iPad. It will always ask for your Apple ID password before a new purchase to make sure that someone else is not purchasing from your account.

If you forgot your Apple ID password, you can reset it at: https://iforgot.apple.com/

Your can change it so that it will not ask you again if you made a purchase within 15 minutes: go to Settings, then 'iTunes &amp; App Store' and then select 'Require After 15 Minutes'.