Pandemic - reduced rule-set for beginners
I don't believe simplified rules exist. For the reference here are the full rules.
This is how we approached it:
First, follow the steps on page 2-3 to setup. Then look at the inset at the bottom right corner of page 3 which says:

Each player turn is divided into 3 parts:

Do 4 actions.
Draw 2 Player cards.
Infect cities.

After a player is done infecting cities, the player on
their left goes next.
Your hand can have City and Event cards in it. City cards are used in some actions and Event cards can be played at any time.

So what we find useful, is to follow these 3 parts in sequence. To perform part 2, follow page 6 &quot;Draw Cards&quot;. For part 3 follow page 6 &quot;Infections&quot;.
Part 1, actions is a bit trickier, because a player needs to be familiar with the actions available to them to make choices. All the actions are described on page 4, with an example on page 5.
In my opinion, good understanding of page 4 by all players, is the most important thing that makes the game following more smoothly, because this is where most variations are. For parts 2 and 3 you can simply follow the script in the rule book - there is not much choice there, and after enough times you'll memorise it and have no problems with it.
By studying page 4 on the other hand, you can understand what options are available to you and make those choices that matter.