I want to put my washing machine and shower stall in the same room on the first floor.
Can I connect my washing machine drain with my shower drain then go to the stack?

Basic answer is yes, but it also depends on code in your area how to pipe this. Where I am at you must run 3" pipe on the horizonal for the washing machine drain and 2" when it turns to the vertical with a 2" trap and stand pipe to 42" above the floor and vented. The shower would be 2" and vented seperatly.