Duels works by holding down R2 slightly to fill the 'dead eye' meter, while this is happening you need to watch your opponent to see what they are doing as they may not draw straight away, dive out the way, pull out a knife or do any number of unexpected things.
The longer you slightly hold R2 down, the more your 'dead eye' bar will fill and the longer your will have to aim. To draw your weapon and begin aiming you need to push R2 all the way down then release the button. When exactly is the best time to do this will vary wildly in every duel. You just need to watch your opponent and draw when you have enough 'dead eye' time to hit the target you want.
Apart from the actions of your opponent, you also need to consider what you want to aim at once you have drawn. If you don't want to shoot to kill you may require a longer time to aim.
Once you have lined up your shot press R2 again to fire.
If you have waited too long to draw and you don't have time to aim before the opponent shots you, then you can press R2 rapidly again after you have drawn. This usually results in a body shot to your opponent, provided they haven't moved out the way, and will usually kill them.
Like said in the comments, and also found in the RDR2 Companion app, you can saddle a horse by pressing L2 to target the horse, and d-pad up te remove and place saddles.
This will be available after the turtorial, in chapter 2. When doing the 'Exit Pursued by a Bruised Ego' mission given by 'Hosea Matthews'. In this mission, the horse and saddle mechanics are explained.
You Can buy and sell Horses at stabled throughout the world. You can break wild horses, or steal owned horses. You can ride a horse temporarily, but to make it yours, you'll need to put your saddle on it.
To saddle a Horse, approach it with a saddle in hand and hold D-pad up to put it on. You can only have one saddle and therefore one main Horse, at a time. To remove a saddle you must approach and hold L2 to lock on and hold D-pad up to remove. You can also saddle a horse by taking it to a Horse Shop, at this point your previous horse will be added to the stables.
Arthur cannot saddle, bond with, or own Horses belonging to other members of Dutch's gang.
Remove your saddle from a dead horse by approaching it and holding D-pad up to remove. Saddles left on dead horses can also be recalled using the stables.
Best Answer
There are some areas like your camp where movement speed is restricted to a rather slow walk. I suspect you were in such an area. Tapping X should work usually, both on foot and on your horse.