To avoid confusion of two separate issues, I'm going to go over both.
First, the person with Initiative always gets the chance to act first, and other players will get their chance in clockwise order from them. So when Deploying, Quickcasting in a Quickcast Phase, and doing creature Action Phases always start with the person with Initiative. Nobody else is doing anything during these times, you're just taking turns, so there are no timing issues.
Second, when things happen simultaneously, such as during the Upkeep Phase or when multiple things trigger at the same time, then each player makes all choices regarding objects and effects they control. If you control multiple objects/effects, you choose for yourself which one happens first, second, etc. If multiple players have things happening simultaneously, and you need to know when they happen in relation to each other, then the player with Initiative gets to decide how they relate to each other.
Example A: In a three player game between Timmy, Johnny, and Spike, Timmy currently has Initiative. Something happens that triggers effects A and B, controlled by Johnny, as well as effects C and D controlled by Spike. Johnny gets to decide the order for effects A and B, and decides that effect B should go off first. Spike gets to decide the order for effects C and D, and decides that C should go off first. So, now we have effects B and C going off simultaneously, and Timmy gets to decide the order for them, deciding that B should go off first. After that we have effects A and C going off simultaneously, and Timmy again gets to decide the order and decides that C should happen. Then we have A and D occurring, and Timmy decides D should go off next. Then we have A occur.
As you can see, Timmy gets to make a lot of choices, but he is limited by the other player's control over their own objects. He could not have possibly made A happen before B, nor could he have made D happen before C. All he can do is decide when you are comparing multiple players to one another.
Voluntary Enchantment Reveals is a special scenario that I'll go over separately. It is sort of a simultaneous event, but due to it's voluntary nature and dealing with hidden information, the rule book makes it very clear that you should handle it in turns. So, starting with the player with Initiative, you may reveal as many enchantments as you want, and then the next player on your left gets to do the same. It continues going around and around the table indefinitely until all players have passed consecutively. Consecutively is important. It is perfectly acceptable for a player to pass on one go around, and then decide to reveal the next time the chance comes back around to them. It's only when all players have passed in a row that the chance to reveal ends. This description could be debatable since the rules were clearly written for 2 player games instead of multiplayer, but that's how I believe it should be interpreted.
This is different from triggered mandatory enchantment reveals, where you don't get a choice on the matter. You can't pass or say you don't want to reveal when dealing with a trigger. You only get to decide what order things happen in if you have multiple things trigger. A triggered mandatory enchantment reveal is entirely under the rules for handling simultaneous events as outlined in Example A.
Example B: After any step, phase, or activation, you should check to see if any player wants to voluntarily reveal an enchantment. If someone does, then you start with the player who has initiative and go around the table taking turns, revealing as many enchantments on your turn as you wish, until everyone passes in a row.
Example C: Something happens to trigger one of your enchantments. You immediately interrupt the action to declare that you are required to reveal something. If you have multiple things that triggered, you choose the order. If multiple players have things that trigger, they all announce this and the person who has Initiative gets to decide which player resolves first, as described in Example A. You do not have to tell the person with Initiative what a triggered enchantment is before resolving it, just that it has triggered.
Note: Mandatory Reveals trigger in response to things happening, meaning that they actually get revealed during a step or phase. You cannot voluntarily reveal other enchantments while handling these triggers, unless the triggered effects provide steps that you can reveal between, such as making an attack.
I hope this helps everything make sense. As you can see, it's actually a very robust system despite it's simple description. Take turns when acting. You always control your own stuff. Initiative decides any other timing questions.