washing machine drain and kitchen sink drain
If an upstairs drain is flowing out of your washing machine drain then the clog is downstream of the washing machine branch drain and it may not be your (or the washer's) fault.

You will need to run a cable (snake) down the pipe to clear it, starting with the drain you think is closest to the clog (might as well start at the washing machine drain, you at least know that it is closer to the clog than the upstairs kitchen drain).

There are a lot of troubleshooting questions and steps a plumber might ask/take, and experience and skill matter a lot for these jobs. I'm not saying you should not try, but calling a plumber is likely to be less frustrating and a lot quicker than doing it yourself if you have no experience with drain cables.

If you want to learn, there are many questions with good answers already on this site, and lots of guidance on your favorite internet video hosting site. My advice would be to purchase a high quality drum auger like this and try it yourself: