Peyote stitch is a fun stitch to do, but I've read so many times on different forums and bead lists that people have a hard time reading and following the patterns. (I did too when I first started.)  
I was making a Peyote stitch bracelet, so I thought this was a good time to share the way I do it. There is no right or wrong way....whichever works the best for you is the best way to do it! (see links at bottom of the page for other methods)
Hopefully this method will make it easier for some of you who have struggled with it. :)
Reading those dang Peyote stitch patterns!
I start off by marking the first set of beads that I will pick up. As you know, this turns into the first and second row.
Stopper bead
Then I number the next row, the 'up' beads (in a different color) in the direction I'm going.
When I'm done with that row, I use red again, and number the next row going back the other way.
And again, when I get to the end of the row, I number the next row going back the other way. (using black again)

I use two colors of pens to mark it. One color for the right to left rows, and another color for the left to right rows.

Since the stopper bead is left on until I finish, it's always easy to see where I left off, and which direction I'm going, even if the piece has been sitting waiting patiently for a week or more. ;)

For instance, in this pattern, I can easily see that I need to start at the blue X and work my way over to the right, because the last 'up' bead marked with a 4 was at the end of the 5th row, and on the left side of the pattern.

Notice too that the numbers on the left to right rows are the same going vertically, (as are the numbers on the right to left rows) this also helps to see where the next row should go if you get lost. Just put in the same numbers as the row under it!  :)   (i.e. Red 4 on top of red 4, black 4 on top of black 4, etc.)
Of course you wouldn't need to number the rows when you're doing a simple 4 bead across pattern like this, but for larger and more complicated patterns it really helps.  It only takes a few seconds to mark the rows, then I can just glance down as I'm pulling the thread through to see what my next color is, so I don't have to stop to look.
If the graph is a simple one, with only a couple of colors, another way to mark off your rows would be just to use arrows showing the direction of the row you're on,  Using the stopper bead for orientation.
Don't fill in the numbers all at once though because this defeats the purpose of knowing where you are in the pattern!  lol
Here is a picture of the way I set up to bead a Peyote stitch piece. I keep the graph under the tray so I can see it easily and still reach my beads.  As I said, this way I can just glance down as I'm pulling my thread through a stitch to see what my next bead is without stopping. :)
That gorgeous pattern I'm working on in this photo is a pattern by Sandra D. Halpenny and can be found here.  (Item number 3565)
Here are a couple of links to other sites that have different methods for reading patterns:  (there are lots more....just do a search on google, and you'll find a bunch.)   :)
About  (see tip #3 in the directions)
btw.... You can start a pattern at the top and work your way down, or go up from the bottom. It doesn't make any's just my personal preference to start at the bottom. :)
Have fun and start doing all those Peyote patterns you've been collecting!  ;)
btw.... if you're wondering why there is a straight line up the right side of the pattern, it's because this is an odd count pattern, and I prefer doing even count Peyote. I will add that column when I'm done, using brick stitch.  It just goes faster for me to do it that way instead of making the turn required with odd count Peyote.  ;)