I don't ride english but I can tell you a break-away halter is just a regular halter with leather or velcro that will "break away" if it is caught on something. Its a safety halter used for turnout. Oh, and I'm sure you will need a saddle pad and possibly some reins. Not all bridles/headstalls come with them.
You'll surely need a saddle pad. If you are riding English or any kind of discipline I'd suggest a broken snaffle for the bit, most come broken but there are the occasional that aren't, D or O ring. I prefer D-Rings myself. I honestly don't see much difference between the D-Ring and O-Ring but some people say there is a huge difference.
As mentioned you would need the stirrup leathers and stirrup irons as well as a girth. If you need to find out your horses girth size, measure their heart girth (from the end of the withers as they meet the back all the way around the horse, keeping the tape where the girth would lie on their chest. View image bellow if you need to.) and divide it by 2 and then subtract 3. Always round up. Most girths come in even numbers so if your horses is like a 33 or 35, you might have to get a 34 or 36 inch girth.
Crops are always useful, you might also want to get a lunge line and some lateral lunging as well, they will help with practicing turning and other things. Normal lunging too often can be stressful on joints and may get boring for the horse, with lateral lunging your horse is free to move around more in any direction rather than in a circle. It's much more stimulating and interesting for both the horse and you.