If you do not own a horse I do not recommed buying a bridal or girth but I do recommend buying a saddle pad and saddle
Fly spray, saddle pad, tack wash, treats, tack trunk
Oh ok, I get it aha ! Sorry :-)
I don't know what you already have in your grooming kit because they can be different from a brand to the other (I guess that you just have the basics), but I think that it is always good to also have hoof oil just in case, a sponge and a sweat scraper to wash the horses when it is hot, a fly repellent, and everything needed to clean and take care of the equipment (especially the leather). I hope this can help, I tried to think about basics that you might not already have.
Then if you already have all this, some things are already nice to have, such as your own saddle pad, your own boots, your own fly veil, that kind of things that allow you to show your personality on the horse aha !
Anais yes I do have a helmet, ect. but what I was asking was tack wise. (I should have clarified)
Do you already everything that concerns security (helmet, etc) ? If not, that is very useful and would save yourself from borrowing it every time you take a lesson.