When I flush my toilet, it goes down very slowly, and then backs up into my shower....will a Draino or Liquid Plumber solve this problem??

I would start by getting a plumber's flat steel snake and work that slowly down the shower drain until you feel the clog, then work the snake back and forth until you feel the snake break up the block and the water flows freely.

Please take the grill off the drain and move it and it's screws into a rag and then place the rag out side of the shower so that nothing goes down the drain but the snake.

When I flush my toilet, it goes down very slowly, and then backs up into my shower....will a Draino or Liquid Plumber solve this problem??
I don't think so!! I am no fan of putting harsh chemicals in your drainage system. Besides, they only seem to work in TV commercials. The only to be sure a drain's cleared is to snake it.
You have a partial blockage in the bathroom main drain.; This can be snaked by removing a toilet and snaking through the open closet bend or through the lavatory roof vent. If you snake from the roof vent put out enough cable to reach the base and 20 feet more. Flush the toilet a few times to flush away anything that's left. Good luck, Tom