She taught me to start with a "thread bunny" or "leader/ender"; basically a small scrap piece of fabric that is folded in half so you have two layers, and is always under your needle before you start sewing. You begin your seam on this scrap and then keep sewing right onto the pieces you are piecing together.
As many of you know, it is so hard to get the seam started right at the edge of the material without sending the material down into the feed dogs/bobbin area. Since you don't back stitch seams like you do with clothing and can thus start a little bit in from the edge, this thread bunny/leader really helps you start before the fabric.
When you finish sewing two pieces together, just feed the next set right under your presser foot and keep going. Sometimes I will sew a whole line of pieces, then clip off the ones behind the current piece, pin them together and just keep going.
At some point you will have to end, and that is where you feed a second piece of fabric under the needle to await the next round. As you can see above, I even chain piece diagonal seams. (If you are wondering about my presser foot, my regular foot is not an accurate 1/4" so I improvised.)
If you are really clever, you can actually use scrap blocks as your leaders and enders so that each time you start sewing, you are also sewing together two scrap blocks. Eventually, you will have sewn enough to make a scrap quilt. I haven't been quite that motivated, but I think it is a great idea!