Just to define the situation a little better.
If you have blood entering your 'fecal stream' from up inside you, your bowel movement will be very dark, almost black.
If you are seeing red blood ON your stool, it is most likely that you are irritating your hemorrhoids.
I have experienced both hemorrhoid irritation, as well as prostrate irritation both as a result of poor saddle fit as well as from wearing cycling shorts. In my own case I found that buying a Brooks saddle (Imperial for me) and wearing loose shorts helped immensely with (most of) the situation. My biggest issue now is the lack of testicular retention that goes away when you aren't wearing cycling specific shorts. I am currently experimenting with various fits of briefs to help alleviate that issue.
My suggestion to you is to seek out a good cycling shop in the area that can assist you with saddle fit. Try the shorts with a good chamois as well, could help. Might also adjust your seating positioning and handlebar height, as you well could be riding with too much of your weight on your backside.
Good luck