There are a number of people that do the quilting by hand, but I honestly don't know of anyone on this board that pieces by hand (except English paper piecing, but that's different). I admire the fact that you want to learn a traditional skill, but hand piecing seems like torture to me! You will never see that sewing!
Taking a how-to-sew class to introduce you to machine sewing is a great idea. I would suggest that you look at vintage sewing machines before buying new; you seem like the kind of person that would appreciate a classic machine, and for quilting you really can get all you need at a very good price this way. People in the vintage section of the board can give you good advice on selecting a machine.
For your current project, if you want to get it done but keep on piecing by hand, consider making it smaller. I don't know how big you were planning on making it, but it could always be sized down to become a throw, or a baby quilt, or a wall hanging, or a table runner... Hand sewing fabric you aren't crazy about isn't nearly as much fun as working with fabric you love. It's always okay to reconsider a project partway through!