Does all come with the saddle - No Tasia.
You'll need stirrup irons which leave a gap each side of the boot.
You'll need stirrup leathers - Aussie synthetics are the best they don't stretch
You'll need a girth strap - again Aussie synthetic works well. They come in sizes.
You need bridle including nose band,
You might need a flash band - this is a strap which holds the bit in the horses mouth- it needs careful adjustment.
You might need in due course a martingale - depends on the horse.
You'll need a bit - try at the beginning a French Link - which is very mild or a standard ringed snaffle. But your tutor will tell you if you need something firmer.
You'll need experienced help to set the tack up correctly. You might have to change it later.
As an experienced Western rider you can get away with a lot without change - but if you are serious about riding English then at the beginning you need to understand the principle of "collection".
Purely as an example for this post: - my horse is forward going - she is always stepping out smartly with impulsion (look it up). I aim for a constant contact (look it up) with her mouth via the bit. As long as I am "in contact", I don't expect her to step out, either lengthen her stride not speed up the pace, without instruction.
When I want her to step out - I let her have some slack
in the reins to allow her to move her head more freely.
If I am to do some tricky riding - say along a bendy, uneven, root strewn path, then I like her "in hand" and "on the bit" - (look it up).
She can be allowed to go "long and low" on a loose rein from time to time to relax but I want her to carry her head high for most of the ride. I want to feel her "impulsion" - the power coming up from her hind quarters
at all times. She is never allowed to dawdle.
My weight I carry on the saddle not so much on the stirrups.
The saddle takes a much smaller surface area on the horse's back - the horse feels more of your movement. Reins are primarily for communication.
First, get to learn to sit properly in the saddle, holding the reins correctly in two hands, right from the very beginning. If your tutor is any good - that is where he/she will start in teaching you.
Having written all this as an Englishman, be aware that American English riding traditions are slightly different from European English - when in America do as the Americans do.