There aren't any simple armies in the Warhammer 40,000 game. Even if you field 4 units of the same one-trick soldiers, there are army rules and enhancements and detachments and stratagems to consider. Simply put, Warhammer 40,000 is a game designed to provide options to the player and force difficult choices. The army you choose, and the list you build for deployment, influences what game you're going to play. The important thing to know is that you don't have to start with all the options turned on. You can play simple 40k for a quick and easy gaming experince, or to learn the game (and introduce variety as you get more comfortable with the system.) Here's how.
Core mechanics are the steps contained in the algorithm of playing a game. When you learn a well-documented game (such as Warhammer 40,000), you learn a procedure. First you do this, then you do that, and finally you do something else. As long as you do all those steps, in that order, then it can be said that you are playing the game. If you don't do those steps, or you add more steps in, then you are not (strictly speaking) playing the game.
Special rules are separate subsystems you can integrate into the core mechanics in such a way that the game's algorithm changes. You are still playing the game, but with situational exceptions and extra options.
To play a simple game of 40k, ignore all special rules. Run your army using only miniature profiles, and even those you might want to ignore most special abilities. Just use the numbers for the miniature itself, its ranged weapon, and its melee weapon.
The core mechanics of Warhammer 40,000 require miniature profiles (number values for all the attributes of a miniature, such as Weapon Skill, Ballistic Skill, Strength, Toughness, Wounds, and so on). You use these numbers to determine what you need to roll on your dice to succeed in some interaction.
For whatever reason, miniature profiles aren't provided in the physical boxes of Warhammer 40,000 model kits. You can purchase a rulebook (or download an official PDF for $0) and then buy a box of miniatures, and you still can't play 40k because your physical miniatures have no numbers associated with them.
For a miniature profile, you must purchase an appropriate army codex, or download the army index, or purchase a White Dwarf that happens to have the miniature profiles you need in it. Regardless of how you obtain the profiles, they come bundled with special rules, some for your army, some defining a detachment, and some as part of the miniature profile.
Obviously you don't need "official" numbers to play any game. It's all made up, so in theory you can assign numbers to miniatures about as well as anybody. In practise, it actually requires you to think about reasonable values that would produce a game that feels at least mostly "balanced" (or else it's surprisingly dull), and to be honest that can be a lot of work. It's why buying codexes and dataslate cards is worth the money. Somebody else has done the hard work for you (not always perfectly, but at least it's a starting point.)
I frequently augment official profiles with custom profiles, because I often want to play a scenario with miniatures that aren't part of the Warhammer 40,000 universe, much less the game. I recently added some monsters from Mansions of Madness into a 40k game. Obviously those monsters didn't have official profiles from Games Workshop. I assigned some numbers to each creature and played. It was the path of least resistance to a quick game with the [unpleasant] surprise of some unique monsters that players normally don't encounter in a 40k game, and it worked great.
An alternate method is to re-theme a miniature to just one or two profiles. I do this a lot with Kill Team, because I play out of the 2018 rulebook, which provides profiles for a lot of, but by no means all, miniatures. When I want to use an Adeptus Arbites squad in a Kill Team (2018) game, I flip through the book, find a profile that seems appropriate to how I envision the unit, and use that.
My friends and I get together for quick games of Warhammer 40,000 and Middle-earth Strategy Battle Game frequently, and nothing slows down a game like pausing during every attack to look up special rules and exceptions-to-the-rules and even basic stat lines.
Instead, when we want to play a quick game, we play a simplified game. While we set up our armies, we discuss the approximate power rating of each ("this unit is my elite squad, these are my scouts, these are my tanks" and so on.) Then before we start playing, we talk stats and either show one another the data cards for each unit, or else we jot the numbers in our heads down on scrap paper. After several games with the same group, we've come to understand what to expect from one another, so we have a sort of vibes-based shorthand.
Running a game using just the numbers, either from your own head or from an "official" miniiature profile, works really well. One would hope so, because that's the core game. Whether Warhammer 40,000 broadcasts it or not, the rulebook plus the miniature profiles is the minimal viable product. If it doesn't work, or if it isn't fun, then the game is broken.
But it's not! It works perfectly. I've played many games with a minimal rule set, usually against players who have literally never seen a wargame before. That was me, until just a few years ago, so I still remember how complex it is to remember how to measure movement, how to memorise the phases in a round, how to decide what to prioritise during a game, and then also, on top of all that, what number means what! Playing with a simplified rule set is complicated enough for new players, and it's still a blast. Some of my most memorable games have been simplified ones, not because the rules were simple, but because the simplified rules meant I could introduce a friend to the game and share the excitement of discovery and mutual [pretend] slaughter together.
Eventually, you might want some variety in how your games play, and that's where special rules come in handy. Special rules are beautifully modular, and I think Games Workshop does themselves and its players a little disservice by failing to treat them that way. I wish special rules were delivered as a deck of cards, some generic and some specific to a certain model or a keyword or a specific detachment, so players could pick a few, or a dozen, or none, before a game.
Maybe you've played a few games with the simplified rules, and you've come to feel that the little mage-looking miniature probably ought to be able to do something magical. You can select a special rule from the army's index, or from an army enhancement, or from a detachment, and implement this new rule. (To make it fair, your opponent's army would get a special rule for a miniature, as well.)
A new special rule, of course, is something extra to remember while you play. Maybe the rule is triggered by something that happens on the battlefield, or maybe it's an ability you can use during a specific phase of your turn, or maybe it prevents something from happening, or boosts an existing number. Whatever it is, it's a thing that's not built into the core mechanics. It happens in addition to the game's normal algorithm. The more special rules you add, the more each miniature can do, which means you have more to keep track of as you play. You can add as many or as few special rules as you feel comfortable with, though, so there's no pressure. You can play a dozen games with the same single special rule until you've internalised it. Then you can add a new one, and play another dozen games. Or maybe you're a fast learner, and you're able to add 3 at a time, with just 2 games between. Or maybe you have a head for rules, and you just add all of them in at once. It's up to you to set the pace, and the good news is that it's never not fun.