My situation was different (as it was green rider + green horse with no formal training combo), but we started showing after 6 months of lessons for my horse and 9 months lessons for myself (I couldn't trailer my horse for lessons in winter, so kept taking some on boarder's horse, which was not nearly a schoolmaster lol!). My goals were for my horse mostly though (as my own experience was just coming together).
Since you ride on lesson horse (that suppose to know their job) and you have 10 years of riding experience yourself, I'd expect you progress faster then we did. I'd say in month or 2 you should learn to feel the contact and start learning correct cues, leg yield, and so on. Sounds like lots of fun to me! Good luck with your lessons!