Decide which you want to do and then do that. Or get a horse for each.
I am going to disagree with this.
I think the key factor here is what LEVEL you want to do all these things at. No, you aren't going to take a horse to win the AQHA world in Western Pleasure, and then next weekend take the horse to the National Finals Rodeo to win barrel racing.
But if you want to go to a local level show and enter all the events with one horse? Nothing wrong with that. And as a matter of fact, it can easily be done.
At one show, here's how my horse Red did:
--4th place halter
--3rd place western pleasure
--3rd place reining
--2nd place competitive trail
--2nd place pole bending
--1st place ring race
--(almost placed in barrels ... 5th place)
There were about 15 adults in entered in each event.
But I do very specific things for each event.
The slow events, I have a low-port solid mouthpiece for my bit. No breastcollar, no sport boots. I make sure my body is relaxed, reins are loose, and give him a calm "waaaaaaalk" voice cue when we enter the arena.
For the speed events, I use a Carol Goosetree bit, barrel reins, sport boots, and a breastcollar. I have some rein contact going into the arena, and my body is "excited". I'm also verbally saying "Ready, Red??? Ready?!?!" in an excited voice.
He darn well knows when it's time to run, and when it's time to look have a pretty WP job.