Ginochka, you can get one Mark-B-Gone blue marker for $3.75
http://columbiariverquilting.com/mark_b_gone.html
You can get one Clover washable blue marker for $4.19
http://www.createforless.com/Clover+Water+Soluble+Marker+Blue+Fine/pid4915.aspx
Or, you can get
eight Crayola WASHABLE markers for $4.49 and they come in different colors:
http://www.staples.com/Crayola-Washable-Markers-Fine-Line-8-Box/product_430942 (Probably even cheaper right now with back-to-school supplies on sale!)
The only draw back to the crayola markers is that you can't "erase" the marks like you can with the other 2 markers. When I mark something wrong, I just get a different color marker and re-draw it.
No matter which of these markers you use, they MUST be washed out of the finished quilt. (Spritzing the blue markers to make it disappear only drives the chemicals into the batting to re-appear later!)
Borders first? It depends. I always stitch my outside edges down to be sure they stay square to the frame, and I will stabilize the borders as I go if I'm doing the center of the quilt first. I like to stabilize the entire quilt first so that I can then roll to any part that I want to work on without worrying about shifting. Sometimes this means that I baste the entire quilt with Superior's VANISH thread before I ever start quilting. That's what I did with "It's a Small World". Then I did the fill in the center and did the borders last.
As for your borders, I would stitch them like I drew them. Stitch around each square and do parallel lines between them. In fact, you could do the same treatment for the next row in, so the unquilted squares would alternate in the parallel lines. I'm not sure how much quilting you are intending to do, so it's hard for me to say.