Keep a notebook in your riding bag and take notes at the end of each lesson, and keep a progress diary with all your riding goals. Discuss your goals with your trainer. Exercise! Exercise is free and you can do it every day 😉 Ask questions, watch other people riding, video your lessons whenever you can. Go to horse shows in your area and watch the riders - not just in the ring, but also watch them warm up and cool off. Take your notebook and write thigs down!
Have a friend come video your lesson so you can see what your trainer is critiquing as he/she is teaching from the ground. Then, review the videos over and over again. It's so easy with smartphones these days. Don't ask your trainer to take the video, ask a friend, family, etc.