Here's good news: with english paper piecing, it's the paper that needs to be cut precisely, not the fabic. Yay! So don't stress this part. Cut some fabric so you can get on to the sewing.
Diamonds are pretty easy, and you get to practice using that 60 degree line on your ruler. Start by cutting your fabric into strips 2 1/2'' by the width of the fabric (though if you don't have the whole width, shorter strips can be cut in the same way).
Lay one strip on your cutting board along a horizontal line, with the folded edge to the right. Put the folded edge halfway between two inch marks. Lay the next strip just above this one, but place its folded end 1 1/2'' farther than the first strip. Repeat with consecutive strips. I was able to cut six strips at once.
Line up the 60 degree line with the bottom of your first fabric strip, so that the ruler hits the bottom edge of the fabric strip on the inch mark, and just cuts off the folded edge at the top. It should align to cut off the folded edge all the way up your stack of strips.
Then move your ruler 3 inches to the left. Make sure that 60 degree line is even with the bottom. You can double check to make sure the ruler is hitting the top strip about three inches from where your last cut landed, for reassurrance that you are holding the correct angle. Make this cut all the way up the stack of strips. Keep going down the strip, making angled cuts every three inches.
I got 12 diamonds from each strip of standard width fabric. Cutting six strips together, that's 72 diamonds at a time. This part was fast and fun for me. I probably cut half of what I need before I just had to start basting them!
Now for cutting hexagons, first you make a cutting template. I just used some scrap paper. Tape one of your paper hexagon templates to the scratch paper and then use your see through ruler to mark a line 3/8'' away from each edge. Now you have a new, larger hexagon for cutting your fabric. You can use this same approach to cutting a single diamond template which is perfect for scraps or fussy cutting.
I laid the cutting template on my fabric and cut out one hexagon with the flat edge just above the selvage edge of the fabric. Then I moved the cutting template up and to the right so it was flush with the hexagon I just cut, like in the picture below. Then back down to cut another along the selvage edge, all the way across the 1 1/4 yard piece of fabric. . I went in horizontal rows back and forth until I had cut out 25 whole hexagons.
I saved all the odd shaped bits in case I can use them for finishing at the end. I also cut five half hexagons, by folding the template in half point to point. Make sure to add a seam allowance along the long edge.
Of course, you can also cut your hexagons one at a time, as you need them.
Feel free to ask questions if these instructions aren't clear! It's my first time doing a big EPP project and so I won't be surprised if I leave something out. (P.S. the main quilt along post is here, with links to all the posts so far)