Book-keeping is not an organizational problem, it is a social problem.
Attitude to book-keeping is generally what makes it a thorn in one's side as opposed to a simple action without great effort required to undertake. This is almost regardless of the level of granularity you use in your games.
Here are some things that help keep people in the right frame of mind when it comes to doing 'sheet stuff' -
Make sure that actions the player takes have thematic effect on the world. People, players not excepted, understand things conceptually before they understand them mathematically. That's why stories remain the most popular medium of data transference. If you want someone to understand something on an instinctual level, like encumbrance, have it matter. Tell them their backpack is dragging them down - they'll shuck it, spending a whole combat turn to do so, which you then reward them with by stopping slowing their speed, whatever penalties encumbrance has. You don't demand they keep track of these things - you simply set up scenarios where these things matter.
In the forest, sniping and running, asking the ranger how many arrows they have and watching them be dumbfounded as they realize that they should be rationing their shots in-character as they response to this ephemeral, multi-pronged effect - as they call out to the fighter that they only have 5 arrows left, it will click for them why it's in the game.
While chasing the goblins down a tunnel, being encumbered and squeezing, you're describing them being 'stuck', asking them for an escape artist roll as the goblin they were chasing turns around with a wicked grin on his face and a sharp, sharp knife - you'll see that dumbfounded look as they suddenly understand what the spacing and squeezing and encumbrance and escape artist numbers on their sheet actually mean.
As they're staring down at their HP total, hearing the cackling of the baatezu jailers as they make their way up the stairs, knowing that behind them is only a dead end where they could only possibly Hide, trying to work out if they have enough hp to survive taking the hits they know they'll take trying to take out these guys with only a length of rusty chain - you'll see that dumbfounded look as they suddenly realize what HP actually is.
Link actions definitively to sheet concepts. Add attack bonus to perform when using a sword to 'show off' in a gladiatorial bout. Use Escape Artist to get out of a mental prison instead of Will. Take a Tumble check to run down the deck of a storm-tossed boat at speed. Use a Climb check to cling to a Hill Giant's shaggy neck-hair as he tries to catch the rapidly fleeing Rogue. Encourage creative use of sheet numbers, instead of letting them be a limiting factor on the possibility space. Of course, don't just let people do whatever - but by allowing more than the listed uses of things, the use of sheet numbers will become apparent (and necessary).
Force players to keep things moving during combat rounds. Having an action ready BEFORE their turn, not rolling a die unless they have the modifier in their head, and doing HP removal or looking up conditions they have just received etc after their turn (with the possibility of missing events unless they do it fast) will force them to intuit the numbers on a much larger basis. Little rules like this will force them to be more natural about their understanding and use of rules and numbers.
Clearly show that you are doing the same with monsters and NPCs. Have Trolls use Str for everything they can, avoiding any skill not based on Con or Str, and acting like it's normal. Have the barman bring his daughter in, as he 'has no head for deals', and his daughter is studying to be a cleric - she has high wisdom, and it's clear. And he thinks it's normal. By binding rules interactions into the setting, you normalize it, and layer expectations for the players interactions with their own sheets and ooc concepts.
Once people have these things conceptually in their heads, ability to perform it quickly and without fuss or questions on their sheet will immediately follow. It will be important to them, so they will remember the things about it. At the same time, it will fade into the background and they will focus more fully on the in-game actions and description - which is good. That's where you want them focusing (or you wouldn't be asking about ways to streamline the mechanics).
You can also apply things in the answers here in terms of optimizing speed of crunch, but ultimately, until people have the links between the description you are doing and the numbers on the piece of paper in front of them, they will not be able to effectively iterate their actions at any sort of speed or level of ease. Practice will do this, on it's own, to a point - but you can ease the transition considerably with little more than making a point of applying the rules fluidly to your own interactions with the game world - something which, as the GM, you are uniquely suited to do - since you control both the mechanics and the world itself.
The actual methods vary from group to group. I vastly prefer 'eyeballing'. Some groups prefer the math. I don't judge. The thing with tricking your brain into understanding things conceptually, though, is, at least in my experience, universal.
– user2754 Jun 17 '14 at 14:25