Romans 8 speaks of the Holy Spirit in relation to the other Persons of the Godhead and the work of Christ and assurance in Him; this is a wonderful chapter to read if studying the theme of the blessing of the Holy Spirit.
Hello @ablacksheep, the folks at GotQuestions.org are very good at answering questions like yours, and I believe that they are also a good place to (at least) start looking into a new topic of interest. Here are a couple of links to their website about, 1. What does it mean to be anointed & 2. What is the new anointing, that may prove to be useful to you.
I've never personally heard the expression "new anointing," but I have moved in Pentecostal circles for most of my Christian life, and the term "anointed" is generally used in reference to ministers, teachings, messages, etc. where the Spirit of God rests upon it/him/her in a noticeable way, and the words spoken manifest evidence of being Spirit-uttered and Spirit-led. This is as opposed to teachings, messages etc. that originate in the flesh, and are coming from men rather than God, or to minsters who are operating in the flesh vs. speaking as the Spirit of God leads them.
That's a rough draft of what the term generally means in Charismatic circles. Outside Pentecostalism you will get somewhat different definitions. But then, they're not usually the ones using the term anyway.
God bless, and keep walking with Christ.
Yours,
Hidden