How many hands high would I need?

If you are interested in horse riding, you will need to take riding lessons and the instructor will choose a horse suitable for your size and riding level so it's not something you need to worry about at this stage.

I am 5'5' and 9 stone so similar in height and weight to you. The two horses I mainly ride are 14.2hh and 15.2hh. They both fit me nicely. You'll likely be put on something 15hh-16hh. Possibly a bit smaller but probably not bigger due to being a beginner. You don't want to be over-horsed as this isn't a good start to gaining your confidence although big horses can be very quiet and gentle too... As long as a horse can carry you comfortably (horses can comfortably carry up to 25% of their own body weight, some ponies can carry up to 40% of their own body weight) and your feet don't dangle below the horse's stomach with the stirrups fixed at the correct length, the horse/pony is not too small for you.

I'm about the same height as you and I ride a 15.1 hand horse... I could go smaller to about 14 hands but 15 is comfortable!

I think your instructor should know that. Don't go out and buy a horse if you don't know enough about what you need, how to care for, or how to train them. If you are asking about your lesson horse, it has much more to do with being the right horse for you rather than size.

Whatever you're comfortable on. You could ride (without hurting) anything over around 12.2hh. I'm around the same height and I prefer something around 15.2hh. Take lessons before buying your own horse though, your trainer will put you on a horse sized appropriately.

If you have real long legs you should have a taller horse. I would think a 15 hand horse would be a good height for you.

15.2 to 16.3 hands high would suit you