How do they work?  I have the same one I started out with or so I think.  I know you can buy more expensive ones in the stores or tame them in the wild but how do you know which ones are faster or have more stamina?  Also how do you keep these new horses without losing them?  Hitching them to a post?  Can you have more than one?
 
Also with loyalty or whatever it's called with your bond with the horse.  When I first started riding it gave me a message that I was building loyalty or something like that with the horse.  How can you check what this is?  I've never seen anything more about it since that first message.
 
Thanks.

In the stores it states how fast they are. 
 
Hitch a horse to a post to keep it safe.  
 
You can have ONE horse. 
 
As for bonding. It only shows that message once. After that you can tell if you are bonded by the blue meter on the bottom left. At first a small section will not be filled. Once you bond that section WILL be filled. 

" @The_Laughing_Man:  Thanks.. so how about the horses that you can tame in the wild?  Do you ever really know how they rank with speed? "

check the deeds in the general store with the Y button...all the horses in the wild are listed with a deed...you can't buy the deed unless you've tamed that horse. if you want to keep a particular horse you like then you can buy the deed for that horse and "use" it anytime you like. That horse then becomes the horse you whistle for.

So far that's what it seems, I have a deed for a black standard American bred horse that i keep using. If yer horse dies you'll get a random horse when you whistle but all you have to do is use the deed in yer inventory and whistle to get that horse again.

" So far that's what it seems, I have a deed for a black standard American bred horse that i keep using. If yer horse dies you'll get a random horse when you whistle but all you have to do is use the deed in yer inventory and whistle to get that horse again. "

once my stallion hit the yellow color and became super, a cougar took his life in one snatch. 
 
I was having some good challenge progress so i got over the stallion :(

" How do they work? "

1. Lasso and break a wild horse.
2. The deed for that type of horse appears at the general store.
3. Buy the deed for the horse you want.
4. Use the deed from the inventory to select that as your active horse.

In the store I have a bunch of deeds that I can buy and a couple that aren't available for whatever reason.  (I figured it was due to me not having tamed them yet)  I have been taming every horse I see and every time it gives me the message 'New items available for purchase'  and it appears that store has more deeds for sale the next time I'm there.
 
The only bad part about this system is that I have no idea which horses I already have tamed and which I haven't.  It would be nice if we knew which ones we should tame whenever we see them running around in the wild.

if you press y (or triangle) to check the info on the deed in the shop it will tell you the colour of the horse. That should tell you which ones you still need to tame. really I only aimed for the top 3 and picked up others out of boredom or during missions/jobs where breaking was the aim anyway.

As someone who just started the game - are the horse differences big enough to make it worth figuring all that out?

" How do they work?  I have the same one I started out with or so I think.  I know you can buy more expensive ones in the stores or tame them in the wild but how do you know which ones are faster or have more stamina?  Also how do you keep these new horses without losing them?  Hitching them to a post?  Can you have more than one?  Also with loyalty or whatever it's called with your bond with the horse.  When I first started riding it gave me a message that I was building loyalty or something like that with the horse.  How can you check what this is?  I've never seen anything more about it since that first message.  Thanks. "

" In the stores it states how fast they are.  Hitch a horse to a post to keep it safe.   You can have ONE horse.  As for bonding. It only shows that message once. After that you can tell if you are bonded by the blue meter on the bottom left. At first a small section will not be filled. Once you bond that section WILL be filled.  "

" I need some help taming the wild horses. I've been working on this mission for hours. I can't seem to get when and what direction I move the left stick when trying to tame the horses. Please Help. "

" @The_Laughing_Man: the whole time or at specific times. Sorry, Im tired of this mission now  "

Breaking horses sucks and herding horses sucks. Thank god you don't have to do it often at all. Also thank you for all the info, tried to hop on a black horse I seen, turns out it belonged to some 

Shoot them in the face.

" Hitch a horse to a post to keep it safe.  "

1) The missions have to be the storyline missions.
Things like bounty missions, saving random people, treasure hunting, etc. do not count towards the 20 missions.
 
Extra tip: 

3) Don't let your horse die.