Whether you sew by hand or machine is a matter of personal preference. There are lots of people who hand piece rather than machine piece. It's up to you.
The quarter inch seam is the traditional seam for quilters. To figure out if you are sewing a true quarter inch seam, take three strips of material that are 2.5 inches wide (by about 4 inches long) and sew them together. The center piece should measure exactly 2 inches wide. If it isn't, adjust as needed. There are feet available for your machine that have an edge that runs along the edge of the fabric to place the needle at the quarter inch spot. You can also adjust your needle side to side, or put a piece of tape or tape a piece of plastic or cardboard to your sewing machine as a guide.
You don't need to iron your squares after cutting, in fact this can distort the fabric. Iron BEFORE cutting, and starching helps the fabric hold it's shape. But you can iron your seams after they are sewn. First, lay the piece down with the two right sides together, just as they were sewn. Put your iron on it in an up and down motion, not side to side, just to set the seam. If you are hand piecing, open the piece up and iron the seam toward the darker fabric. If you are machine piecing, you can either do that, or iron the seam open. Either way, you "press", not "scrub" with the iron.
As far as sewing squares together, are you talking about making a nine patch? If that is what you are making, sew three rows of three squares. Press the top and bottom rows of seams in one direction, and press the middle row of squares in the opposite direction. Then when you place the rows together, the seams will "butt up" against one another nicely. Pin every seam well before you sew.
Good luck and make sure and post photos for us!