Introduction 
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Why are neat schematics important?
At the very heart of any design is the documentation.  The documentation serves many purposes.  It is often created by the design engineer.  It is consumed by the customer.  Which customer?  That is an excellent question. Consumers of documentation start with the design engineer, but he or she must recognize that many other people will also view and use that document to do their jobs.  Manufacturing will use it during the build process, technicians will use it to debug the product on the line.  Other technicians will use it for repairs on returned devices. An engineering team will use it to design the next revision of the device.
For the schematic to be useful, it must clearly and easily convey information about how the circuit works.  I have seen manufacturing struggle with documentation because the engineer made that documentation for himself rather than considering the needs of the consumer.  My first piece of advice is to have your customer review your documentation.  You, the engineer, are providing the schematic for them and it should be useful to them.  Change a color, add information, move something around so that it is easier to understand.
As long as the netlist is right its fine
I have seen a lot of schematics through the years and it seems that some people make schematics only so that the netlist can be generated and a PCB created.  While that is one of the functions of the schematic, it is not the only one.  Years ago, I work for a giant of a company that had more rules than you could shake a stick at.  They had a device that had been manufactured for over a decade and I had to become familiar with it for a cost reduction initiative.  Because this was an assembly with cables and multiple boards and devices, there was a pile of schematics on my desk and I could not make heads or tails of what the designer wanted.  As a tool for myself, I made a large schematic on C sized paper with the entire circuit displayed.  While I was criticized by other engineers, because a single sheet should not contain an assembly, I found that schematic diagram began being copied and distributed throughout the facility.  Why was it popular?  Simple, it conveyed information in an easy to understand way and thereby made it easier for people to do their jobs.
Never place a wet mop in the closet upside down! 
What makes a schematic easy to read is largely a matter of style, never-the-less I am going to give some specific suggestions that have helped me, your mileage may vary.  The most important suggestion is to be consistent.  If you do things a certain way, then try to always do it that way unless there is a very good reason to vary.
Place positive voltage bars up and negative and ground facing down.  If there is a ground on the top of the chip it is very tempting to put the ground upside down.  It is correct but harder to accept visually.  I usually make the symbol so that this situation does not arise.
A section of schematic will be easier to understand if all the grounds and all the power are at the same level.  It can look a little silly if you have to add 2 inches of line to get a ground at the same level as all the other grounds, but it is visually obvious what you mean when you have a signal run down there.
Always connect a pin to something else using a wire.  For example, don’t connect the power symbol directly to a chip, use a short segment of wire/trace in between them.  It makes the drawing just a little bit bigger but I think it is neater and therefor easier to understand.
Choose a grid and stick with it.  I use a grid of 10.  I have set up my library so that pins always land on the grid.  This is important for visual consistency.  The eye will pick up changes in grid size and the overall schematic will look sloppy, a distraction.  Also, off grid object may not actually be connected the way you think.  I often place text using a grid of 5, but I don’t use a grid of 1 because then it will be offset by 4 in one spot and 6 in another and look sloppy.  And sloppy schematics are harder to read and will distract the consumer.  They are less efficient.
It used to be that all schematics were in black and white.  Today we have color monitors and color printers.  A well-chosen color code on your schematics will help convey information quickly and efficiently. I use yellow in the body of a rectangle to portray a chip.  I use the same shade of yellow and the same shade of blue for the lines each time.
I find it much easier to have the negative input of an op amp on top and the positive on the bottom.  It may make your drawing more compact to draw it with the negative on the bottom or vertical up or down, but it is harder for the first-time viewer to digest.
It is not necessary to have everything on a single page.  Group like things together and place them on a separate page.
In general, inputs go on the left and outputs go on the right.  You don’t have to do this exclusively, however, it is much easier to follow whenever you can do it this way.
Use notes on your schematic.  You can hide notes when you print the schematics, or choose to include them.  Placing notes on your schematic will allow you to communicate not only how a circuit works, but some of the design considerations you made when you did the work.
The list could go on and on.  Make a list of your do’s and don’ts if that helps you.  While style is a matter of opinion, easy to understand schematics should not be.  When someone looks at a schematic and says, “That was made by so and so.  I recognize their style,” strive to have that be a positive statement and not a criticism.
Final thoughts
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