Riding tack and cloths
Before considering to buy any equipment, you really have to keep in mind that horsemanship should not depend on tack or gimmick. Ultimately, you should be able to ride your horse bareback and bridleless _ which I do not recommend though as it does not make much sense!
My considerations about tack are limited to the necessary equipment to ride a horse safely and comfortably.
Saddle: Western saddle, Australian saddle, English saddle
What saddle is best? There is no simple answer to such a question: for what? for whom? It would make no sense to advise a saddle against another generally speaking. As it will depend on your practicing western riding, jumping, trail riding, hunting, etc.
Horsemanship concerns ALL riding disciplines, all breeds, all riders. I have already used western saddles, Australian stock saddles, Spanish and classic English saddles, Kyrgyz and Russian saddles. My personal favorite is an all-round western saddle as it fits most horses and is comfortable and stable enough to start colts or spoiled older horses.
There is actually only one important advice to follow: make sure the saddle fits your horse correctly. Otherwise, it will be painful for your horse first, and eventually for you as it will buck you off sooner or later.
Bridle: Snaffle bit, Mecate or McCarty
Whatever the solution you choose (rope halter, side-pull, snaffle bit, hackamore, shank bit), always keep in mind that you have the responsibility to remain light all the time. A heavy hand on a rope halter will be harmful to a horse’s muzzle when a light hand on a bit will be nice and comfortable for the horse.
My personal favorite is a thin and plain egg-butt snaffle bit: thin because I want to make sure my horse has enough space to put its tongue comfortably under the bit, plain because I do not want to spoil my horse’s mouth with squares and twists and who-knows-whats, egg-butt because I do not want my horse’s mouth corners to get pinched by the rings.
Equiknowlogy 101 by Smokie Brannaman
Every horse is capable of doing all the moves we are used to teach them: martingales, draw reins, tie downs, etc. should be avoided for the good of the horse.
Very wide variety of choices, so a very short answer: do as you like. Unless you ride in a specific discipline that requires a uniform (western, jumping, etc.) you should just select what fits you best. Just one advice: aesthetic is not much of importance once you are flying over your horse’s head, comfort and security are the priorities.
If you start riding or practicing horsemanship, please make sure you wear a helmet. Later on, when you feel safe and confident on your horse, you can choose to take it off.