Ask your instructor to cover basic horsemanship as part of your lessons. That's what I do as an instructor. I teach my students how to (safely!!) catch and halter, groom, tack up, horse psychology and reading horses (horse sense). All of these things prepare you for leasing or owning. I absolutely think you should lease before you buy, and that you should still be taking lessons when you do start to lease.
How long have you been taking lessons? I would estimate, depending on the rider, it would take at least 6 months of weekly lessons to be ready to lease a horse.
You could also ask you instructor if you could come to your lesson early to get your horse ready, groom/tack up. Etc.
Ask if you can come watch during feeding time or the next time the farrier comes out to do hooves.
Volunteer to help, even if it is just sweeping aisles, cleaning stalls, etc.
Most of the time involved in owning a horse is in the day-to-day care, not the riding.