8" Dain in garage floor under house and 2" drain in the basement floor do not back up, but I get water on the floor from under a basement wall. I have run water from various locations and can hear it passing through the cleanout connection in the lateral. I suspect some partial blockage is the problem.What do I need to due?

I suspect that you have two issues here. One issue would be the blockage in the stack located in your wall. The other would be the leak letting water out on your floor. Your solution would be to first open up the wall and addressed the leak.
You may now proceed to snake the pipe and clear the blockage. More questions? I'm as close as a click. Good luck, Tom