I would like to add just a few suggestions to the advice given so far (all excellent suggestions!)
1. Before you cut, spray starch your fabrics. I like MaryEllen's Best Press, but you may use any spray starch you like, even if you make your own solution. Just use it.
Try to make your fabrics feel like the weight of construction paper, and then cut them. That's how stiff they should be. I know it sounds like a lot, but I learned that suggestion from Debra Wagner, who is an award winning quilter, and she is right. So much easier to cut the fabrics, and to stack them together and piece.
2. If you feel like you need to draw a seam line (probably not necessary, but if you need "training wheels" until you get the feel of things, ok), place your fabric on a sandpaper covered mat, cardboard, whatever, and then draw lightly with a fine mechanical pencil so as not to stretch the fabrics out of shape.
3. I have used this technique when teaching kids to sew, and some adults can benefit from it as well....
when you are getting ready to place your fabric pieces Right Sides Together (RST), put a tiny drop of elmers school glue (or applique glue...whatever you have) in the seam allowance.
When i say tiny, i mean tiny...like the size of a quilt pin head, or bead.
Do it in a few places along the seam allowance of the right side of one of the fabrics. Then put your next piece of fabric on top to make your little fabric sandwich that needs to be sewn RST. Heat set the glue by pressing the fabrics with your iron. This eliminates the need for any pins.
You can assembly line sew these pieces - just stand at your ironing mat for a bit and glue and heat set a whole bunch, then feed them into the machine and cut threads after they come out the other side of your needle - typical "chain piecing".
You should have accurate 1/4" seams, and the pieces should be exactly one on top of the other with edges aligned.
4. Lastly, it's my opinion...(i'm not a machine tech, so i could be wrong, but i swear this is the case) that some machines "kick" the fabrics out of alignment. It's as if the feed dogs are mis-aligned. I can sew on some of my students machines and i can't sew straight either, then sit down at my machine and i am fine. sometimes it is not the sewer but the machine. if all else fails, get yours checked out, and/or try sewing on some other machines to see if there is a difference.
Hope this helps. Love all the suggestions here!!