Once you've decided to machine quilt your quilt, you will pin baste it, to keep all the layers in place, so they don't shift while you're sewing. What's really helpful, for this kind of quilting, is this little attachment. It's called a walking foot, or an even feed foot. And what it does is, it feeds all three layers of your quilt evenly. That helps not get you any puckers, or wrinkles in the back.
It, basically, your walking foot is good for straight lines. You wouldn't really want to do curves, or complicated designs. It's great for straight lines, stitch in the ditch, that kind of thing.
And you can just sew off your quilt, because what you're gonna do after it's all quilted is, you're gonna just trim off the edges, and bind it, so you won't see anything. I would do the next row. You can either measure it, and decide you want to use a piece of masking tape to write with, say two inches, and stitch it, or you can eyeball it, or you can use a guide that comes, usually, with a walking foot.
Straight line machine quilting is the fastest, easiest way to quilt a quilt. You don't need a special machine. A regular sewing machine will work just fine. It does help to have a walking foot, but, if you're looking for speed, you can't beat it.