HOW DO YOU CHOOSE BACKING FABRIC?
The color of your backing fabric should be of a similar value to the thread you want to use to quilt with. In machine quilting the same color thread is used in top and bobbin so you'll want to make sure the backing fabric supports this. A busy backing fabric will draw less attention to thread starts/stops and anchoring stitches which are part of machine quilting.
HOW DO YOU PIECE BACKING IF THE FABRIC ISN'T AS WIDE AS YOUR QUILT?
If your backing fabric is directional you will need to piece in the appropriate direction. If you are piecing lengthwise then you should measure the width of your quilt, add 8" and figure out how many widths of fabric you'll need to cover this distance. If you need 2 widths of fabric you can either make 2 equal-sized pieces or you can use one piece the full width of the fabric and then add two equal size pieces to either side. Make sure you remove selvages from the backing pieces and seam with 1/2" seams. I like to seam my backing with the selvages on and taking a 1" seam then I snip and rip the selvages off. Press open. Use a similar process to piece width-wise.
HOW DO YOU SQUARE YOUR BACKING FABRIC?
The backing fabric needs to be square for it to mount correctly on the quilt frame. To square the backing fabric, fold it in half lengthwise aligning the cut edges. Shift the cut edges until the fold is smooth. This shift may cause the ends to be out of alignment. Fold the backing in half lengthwise again and again line up the long edges and make sure the fold is smooth. Use your ruler and rotary cutter to cut the ends of the backing square with the fold.