In this day and age, the simplest way by far to move up the scale on proofreading and spelling is to use the software available on most word processing programs. The will check grammar and spelling as you type, or you can wait until you're finished and do it all at once. Either way, the software will offer many excellent suggestions on how to fix the problems.
The weakness of this software is that if you are writing technical information, it probably won't have those words in the dictionary until you customize it. Also, the grammar and style side of the software won't like any innovative or accent oriented departures from normal writing patterns. Very few of these programs can be customized to accept other styles of writing.
In the other direction, proofreading and spelling can be improved without automation. If you struggle with spelling, make a list of the words that you misspell the most often. Work with it like the old fashioned grade school spelling tests until you master the words. Once you've learned to spell your 20 or 30 worst case words, make a new list and keep going until you have built up a stronger spelling vocabulary.
This will also improve your writing. You won't be afraid to use to words for fear of spelling them incorrectly. It will also build up your vocabulary because there are infrequently used words that will be fresh in your mind to add more creativity to what you write.
When you proofread for spelling, read slowly and concentrate on each individual word rather than groups of words. This will force you to actually check how each word is spelled. To catch all of the spelling errors, you will need to read the material about once a day for 3 or 4 days. If this amount of time is acceptable, get several other people to proof your work for spelling errors.
Grammatical errors are more difficult to fix. If your grammar is strong and you were just writing rapidly as you created text, these errors can usually be spotted by, again, reading through the text slowly or even out loud. Try to read each sentence as a stand alone piece. Look for the presence of a subject and verb. Make sure that they agree.
If you have exceptionally long sentences, try to find ways to break them up. This will lessen the likelihood that you have missed the punctuation. It will also help with the subject and verb agreement.
Once you have corrected the grammar and punctuation, take another pass through the material and read each paragraph as a stand alone entity. Make sure that you have maintained the same tense throughout the paragraph. Look for possible problems with ambiguous pronouns. Watch for changes in topic or emphasis within the paragraph that might indicate the need of an additional paragraph break or a misplaced sentence.
Once you feel that you have the major problems corrected within the material, read it a final time for flow. You should see continuity between paragraphs. If the topic makes a dramatic move, make sure that you have transmitted the coming changes to the reader.
When you are happy with the paper, have a friend who is a good writer give it one more look over. Be careful here because some people can't tell the difference between editing and changing the style. Evaluate each suggested change on the basis of whether it is a correction, or someone trying to make the paper sound like he or she wrote it instead of you. A good editor can usually make corrections and maintain the style of the original author.