If it were me, I would ask my trainer if I could clean up and condition the bridle of my lesson horse, and save up for a bridle that belongs to my own horse, or lease horse.
I've heard Plymouth is good for the price, and I have a Suffolk laying around somewhere, and it's decent enough to get the job done, and looks nice.
But think-If you give your lesson horse's bridle a good cleaning and soak in some oil, then you can in the meantime keep saving up for a bridle of better quality.
My Suffolk rarely gets used, and I'm concerned about how long it would last with often use. It already looks a little rough with the small amount of use it gets. Typical cheaper leather. Doesn't last you near as long, no matter what kind of care you give it.
As far as bits, consult with your trainer on that.
"You're just as sane as I am."~Luna Lovegood.