You don't really need to bother. I didn't find any real advantage to getting any upgrades for horse tack or for different breeds over cosmetic differences.
I never really had any situation where I felt that a faster horse might have been better for me.
For stamina, learn to voice-pat. That same button you use to pat/bond with the horse at slow speeds, you can also use at higher speeds - you should hear Arthur/John say something encouraging and the horses stamina metre should increase a bit. Get the rhythm right and you can gallop from St Annesburg to Blackwater without stopping or slowing.