I'm going to go out on a limb and say that in addition to basic high school chemistry concepts, 3-5 years of intensive, immersive learning (i.e. being in a working synthesis lab) is enough to turn you into a competent enough chemist, both in terms of physical skills (choosing and setting up apparatus correctly, being attentive and taking appropriate notes and measurements, handling chemicals safely and efficiently, being diligent and careful, conducting analyses, troubleshooting equipment) and the book knowledge (lab safety and protocol, organic reactions, appropriate conditions, being able to understand literature (and generalize from them), and most importantly, how and where to look up documents to fill gaps in your knowledge).
Even 1 year of lab experience (with supervision) is enough for an aspirant chemist to be able to follow a synthesis and do it themselves. Sort of like how it doesn't take much experience to bake a cake, but to come up with a cake recipe from scratch takes a bit more learning.
At that point you should be able to (possibly,
with some help from your peers) look at a molecule with an existing synthesis, think of an analogue (say, adding an extra group), and plan a synthesis (usually by modification of an existing, proven method - or if you're lucky it's already been done before), and, assuming you can acquire the precursors somehow, carry out the synthesis. If it's a simple compound, you should even be able to plan a synthesis from the ground up, although it may not work well or at all - but you should be able (if you have the right analytical tools) to figure out where it went wrong, and why, and adapt accordingly. (Did you know you can buy a GC-FID off of eBay for less than US$2000?)
I don't think it takes 15 years or anything like that. Maybe if you're doing it all while stoned, or as a part-time weekend job. But if you're serious about it, you can do it in much less.
