How To Solve Sudoku
At first, it took me long long time to do them. Even the easiest one, it would still take me an hour to do. I started to think like a computer, and made a grid within each square and used it as a strike out list. Using the process of elimination, I would uncover by logic what it had to be. This did work, but the process was slow to set up.
My goal then shifted to solving the puzzles with some speed. I switched from making a list of what are the possible number candidates as opposed to listing the ones it couldn't be. This greatly helped me speed up and completion times to under 20 minutes even for the slightly harder puzzles.
I am now currently at the "hurdle" of medium hard puzzles. I am finding that only the Thursday and Friday and weekend puzzles are the only one that offer any challenge. Anything less than 4 stars is too easy.
So now I must learn more rules and tricks about Sudoku to progress further. But before I do that, I will write down what I do know. Now I have been to many sites, and some of them with fancy web stuff like www.sudoku.com actually makes it easy to figure out what the hell they are trying to explain. But there are some, I just can't see their logic. It is not that I think their page is bad, but they could sure use a little more illustration to help explain to the layman how their logic works. So thus, with my limited web page building skills, I will attempt to illustrate what my logic are and hopefully you will find them useful to you.
LESSON 1  Given Rules  
a. Row 
When a number like 1 appears in a row, you can use Row Elimination to remove all occurances of 1 from that row because there is only allowed to be 1 of each of the numbers 19 per row.
Alas, the number 1 is NOT allowed to be in any of these squares in the row. 

b. Column

This is the same as part a above but now applied to a column. When a number like 1 appears in a row, you can use column elimination to remove all occurances of 1 from that column because there is only allowed to be 1 of each of the numbers 19 per column.
Alas, the number 1 is NOT allowed to be in any of these squares in the column. 

c. Block

This is the same as parts a and b above but now applied to a block. This is the last of the 3 elementary rules of Sudoku. When a number like 1 appears in a block, you can use block elimination to remove all occurances of 1 from that block because there is only allowed to be 1 of each of the numbers 19 per block.
Thus the number 1, may NOT be in these squares. 

Ok, this covers the very basic rules of Sudoku. Now some of the elementary solving techniques.  
LESSON 2  Two Line Elimination  
Now that we learned 1a) Row elimination, now we apply this rule for two rows and we will get by process of elimination where the third row is.  
a. Row

What I do is start at the number 1, and look for any two 1's that are in same row of 3 blocks, for example like the one above. After I finish 1s, go on to 2's, 3's and etc.. In this example, we are given where the 1's are in block 1 and block 2, so we need to figure out where in block 3 the one. So we begin by applying row elimination like we did in part 1a, to figure out where 1 is not allowed to be.
Alas, the number 1 is NOT allowed to be in any of these squares marked by "X" and "x". This means the 1 can only be in ONE of the squares marked with the stars. Which one? Well, in this example, it could be any of them, so what is done is something called "pencilling". As you can guess, it basically means you will use a pencil (as opposed to a pen) to write the number 1 in these three squares marked with the "*" as a mental note that the 1 may only be allowed in these three cells. 

b. Column

This is the same as part 2a above but now applied to a column.
When a number like 1 appears in a column, you can use the 2 columns elimination just liek you did for rows to remove all occurances of 1 from those columns because there is only allowed to be 1 of each of the numbers 19 per column. Thus you will be able to eliminate the squares in third block where 1 is NOT allowed to be.
Alas, we eliminated 6 of the squares in block 3 with "X" and "x" where 1's are NOT allowed to be. There is already a 2 and a 4 taking up another 2 squares, thus the last square, (6 squares + 2 squares = 8 out of 9 squares), is the only place left for the 1 to go. So you can PEN in your 1 there.


Now just using these two techniques, I have found I can complete just about any one star or Monday's (maybe even Tuesday's) newspaper puzzle.  
LESSON 3  Cross Hatching  
I really don't know the technical name for this but since I saw some other web site use this term, I will use it too. Now that we learned row and column elimination from the two lessons, now we apply the rules for a column and a row at the same time.  
a. Basic

Ok, the principle behind cross hatching is the same as what you did in Lesson 2, but instead of using 2 rows, we now will use 1 row and 1 column.  


In this example I will start with the 1's. Using what we learned in lesson 2, we will extend the rows and the columns of the 1's in order to determine where the 1's CANNOT be.
So after eliminating all the 1's both rows and columns, you can quickly see that have the two blocks where the columns and rows intersect have some clues.
Block 3 here, has eliminated 5 of the squares. With the 2,5,7 already in the block, that only leaves one square left. You can PEN in the 1 in that last square.
Ok, we also unravelled some information about block 4.
We eliminated 5 of the squares where 1's are NOT allowed. The 7 and 8 take up the other 2 squares, leaving only 2 squares left where 1's MUST be. Now, there are two possiblities and at this moment, that 1 could be in either of them. So what we will do is in PENCIL, write 1's in these two squares.
I will also make a side note, since these two 1's are the ONLY possible locations, I will pencil them in and circle the 1's. They may become useful in the future when we start hunting down pairs. 

b. Extended 
Usually when you apply the basic cross hatching, a chain reaction might occur. So thus, the next part of the lesson is to extend what you learned from cross hatching.
You will remember from part a you managed to PEN the 1 in block 3, I have marked it in green. So I will check to see if what I penned in actually causes a chain reaction and allows me to learn more information about other blocks.
Well, we can quickly see that by applying the 2 Line Column rule, we can pencil in 1's in block 9
Alas, so. So now lets look at the other side of the table. Now, you can see that by the two pencilled 1's (in red) that we are guaranteed that the two 1's MUST be in one of these two squares. Remember that we pencil circled these in the previous example to make them easier to see. Therefore, the 1 CANNOT be in any of the other 7 squares in that column either. Actually if you really wanna push it, the 1's also can't be in any of the other 7 squares in that block either.
So we can start eliminating the squares where 1 is NOT allowed to be.
Well, what do you know. We managed to logically see that in block 7, there is only one place a "1" can fit. So we PEN that 1 into that square (I will mark it in green).
Remember, we were talking about chain reactions?
We look at the bottom row that we just worked on. On the left is a PEN "1" and on the right is two PENCIL 1's (with penciled circles from the example above the 1's can ONLY be in these 2 squares). By using row elimination we can remove one of the red 1's.
Thus, the "1" can now ONLY be in the one spot. So we can change the PENCILed "1" into a PENed "1". As an option, you use an eraser to wipe out the pencil "1" as it is not needed anymore.


Ok, with the knowledge that you have now, you now should be able to do the Tuesday maybe even Wednesday puzzles. You should be able to tackle those puzzles with a few stars, maybe up to a three. But you will need more tricks to your arsenal to do the harder ones.  
Examples  
Basic


Advanced

Ok, here is a puzzle out of a Wednesday Newspaper. This puzzle can be solved with the three Lessons you learned above. Start by using two line reduction and cross hatching starting with the number 1. When you reach 9 you will start a set of chain reactions using just the basic rules of only 19 in any row, column or block to fill in the rest. There are 29 known squares out of 81.
From "The Metro" Wednesday, July 5, 2006. 

Advanced
Paired row elimination 
Ok, I will show how using circled candidates will help eliminate possibilities. This is a 5 star puzzle from the Province 1/1/4/2007.
I will direct your attention to analyzing where the 8's are suppose to be. 