That’s why I myself think John is a good start, because I think that with a lot of the text in it, you don’t need to be as familiar with the OT (like in Matthew), but you still get everything so to speak: miracles or signs, Jesus’ divinity, his humanity, the declaration of St. Thomas at the end, the high priestly prayers of Jesus etc. I could be wrong of course and could easily change my mind if there was a good argument for another book.
Btw, I remember that I once heard Scott Hahn say he would think the Gospel of John is a good start. I don’t know if he was talking about the whole bible or just in reference to the NT. Anyway, for a new reader I probably would only recommend that Genesis be read first, but from there it can get very complicated and laborious.
Finally to convince a non-believer, wouldnt you recommend John anyway to prove that first of all Jesus is God and the Christ? Because the synoptics are not very clear in expressing the divinity of Jesus. (I find that Matthew does provide strong hints in his Gospel though).