One thing that's always bugged me about the horses in minecraft is that they don't feel anything like riding a horse. A big part of that is the way they rigidly follow your mouse movements and do not have varied speed while moving - they feel like they have no weight and are just an extension of the player. They feel even more rigid than walking because you can at least sprint while walking to modify your speed.

I propose that horses should speed up or slow down by repeatedly pressing the sprint or backward buttons. When you get on the horse it will walk, press sprint once to make it run, and press it twice to make it sprint. You can also use the backward button ("S") to slow the horse down one level without stopping. While running or sprinting if you release the forward input ("W") the horse will quickly slow to a stop (but not instantly).

Steering the horse should also be less rigid. There should be smoothing between the player looking to the side and the horse turning so it doesn't follow your mouse movements 1:1. This will feel much more like steering a horse. Basically the horse shouldn't be able to turn instantly, so the player will look where they want to go and the horse will smoothly turn the same way. Also, if you walk diagonally (W + A/D) then the horse should turn and point diagonally while the camera remains looking straight, the same as when walking on foot.

Essentially controlling the horse should feel a lot smoother than it currently does because horses have weight. Making these changes would also make controlling the horse more like regular walking because of the camera changes and the ability to sprint. Ideally this would all be configurable as well for those who prefer more snappy controls.

Actually now that I think about it, a better way to slow down without stopping would be to just let go of W briefly and the horse will slow down one stage every 0.3 seconds or so, so if you take your finger off W for less than 0.3 seconds the horse will only slow one stage, or if you let go for longer the horse will keep slowing till it stops after about 1 second from a full sprint. If you need to slow down faster you could hit reverse (S) and the horse would stop immediately and rear up. This would work better than tapping S to slow down because it turns out it's difficult to hit S while holding down W.