Many people here are recommending texts like How to Prove It by Velleman. From this book: "Many students get their first exposure to mathematical proofs in a high school course on geometry. Unfortunately, students in high school geometry are usually taught to think of a proof as a numbered list of statements and reasons, a view of proofs that is too restrictive to be very useful."
One solution to this problem is to learn from a book purely on proofs, as Velleman suggests. It seems to me, however, that given your background and your wish for an 'exceedingly gentle introduction', this method might not be the best for you.
While I agree with the view that the attachment between geometry and proofs is detrimental to students' learning of both topics, I would like to make the argument that in a case like yours, learning proofs through geometry is actually a great place to start.
Proofs are not traditionally linked to geometry without reason; in school, geometry is the closest thing you get to "real" mathematical thinking. Since you are already familiar with geometry, revisiting it, this time through a lens focused on proofs, would be an effective way to bridge the gap between traditional school mathematics and proof-based thinking.
At this point, it comes down to finding the right geometry book. I highly recommend Introduction to Geometry by Richard Rusczyk: https://artofproblemsolving.com/store/book/intro-geometry. While it is designed for the advanced high school student seeking to learn geometry in a different way than what is taught in school, it just so happens that this makes it a great book for your purpose as well.
Having already learned geometry, you will be able to focus more exclusively on the proof aspect of the book. Take a look at some of the excerpts listed in the link above to see if the style of the book suits what you are looking for. After learning geometric proofs, you will then be able to easily extend the same ideas to proofs in other subjects.
One solution to this problem is to learn from a book purely on proofs, as Velleman suggests. It seems to me, however, that given your background and your wish for an 'exceedingly gentle introduction', this method might not be the best for you.
While I agree with the view that the attachment between geometry and proofs is detrimental to students' learning of both topics, I would like to make the argument that in a case like yours, learning proofs through geometry is actually a great place to start.
Proofs are not traditionally linked to geometry without reason; in school, geometry is the closest thing you get to "real" mathematical thinking. Since you are already familiar with geometry, revisiting it, this time through a lens focused on proofs, would be an effective way to bridge the gap between traditional school mathematics and proof-based thinking.
At this point, it comes down to finding the right geometry book. I highly recommend Introduction to Geometry by Richard Rusczyk: https://artofproblemsolving.com/store/book/intro-geometry. While it is designed for the advanced high school student seeking to learn geometry in a different way than what is taught in school, it just so happens that this makes it a great book for your purpose as well.
Having already learned geometry, you will be able to focus more exclusively on the proof aspect of the book. Take a look at some of the excerpts listed in the link above to see if the style of the book suits what you are looking for. After learning geometric proofs, you will then be able to easily extend the same ideas to proofs in other subjects.