Why is the upstairs sinks and shower water draining into the downstairs shower?

Because there is a blockage in the main drain downstream of the downstrairs shower. It would like to come out of your downstrairs sink too but the shower is lower and water like downhill. You need to rod your drain. Do you have a clean out in the basement or crawl? Back to you.

Ok ballengerb1. We had a plumber out about a month ago, who did just that and it is still happening. Our clean out is in the basement behind our washer and dryer. Anyway, he told us that our tree in the front (it's a big tree) may have preforated the piping with its roots and that may be our problem.

Going in, let me say that sooner or later you're going to have to bite the bullet and replace that old sewer line with PVC.
RootX or Robics Foaming Root Killer that contain Dichlobrnic are two products you could try.
A less expensive way would be to call around to garden supply stores and ask for fine grain Copper Sulphate. Put a 1/2 pound in your toilet and flush it down. Repeat in 6 months.
Hope this helps Tom