When I was at "Junior" school (age 7 to 11) in the UK, we had "RK" (Religious Knowledge) classes, which were more or less New Testament stuff, related to Christian festivals such as Easter, Christmas, Palm Sunday, etc. It can't have gone into much detail because I seem to recall not knowing what a "virgin" was until I was about 12. At a private ("public") secondary school (11 to 18) we had "Divinity" classes conducted by the School Chaplain, and these were resolutely non-controversial: no Holy Ghost stuff, not a lot about Jesus, mainly Old Testament history & Saul, Nebuchadnezzar etc. All very Anglican. In both cases I see now that it was perfunctory in the extreme, and the message was not "this is the truth and it's majorly important", rather "This is what it says in the Bible and we've got to teach it for an hour every week because the law says so, but it doesn't really matter".
As for the original question, I personally imagine that "a new world based on the fruits & gifts of the holy spirit" would probably be rather like what you see on the cover of the Watchtower
. That is, a free picnic with zoo animals and a conservative dress code. Lots of smiling, although I think Donald Trump wouldn't be happy, judging by the picture below. There will be hairdressers who learned their craft in the 1920s. There are plenty of discussions of the "World to Come" in the Bible, and a cursory Google search will bring up plenty of ideas- lions laying down with lambs, etc. See the Nicene Creed, also Matthew 12:32, Mark 10:30, Luke 18:30, Hebrews 2:5, Hebrews 6:5. Other related expressions are "age to come" which is typically found in more recent translations, Kingdom of God, Messianic Age, Millennial Age, The New Earth and New Jerusalem, and dispensation of the fulness of times and possibly also eternal life.