I have a home built in 1965. I have galvanized piping as drain pipes. My washing machine drain pipe overflows every time I run a wash cycle. I have shoved a garden hose down the drain and run it full blast and it does not overflow. This leads me to believe the drain has narrowed due to corrosion and is not completely blocked. I think the pressure and amount of water from the washing machine is just too much for the drain. I have not removed the drywall yet to see exactly what is going on inside the wall. I want to replace the pipe. I want to install another drain and water lines for a utility sink while I am at it. The pipe appears to go straight into the foundation and up to the roof for a vent. I am wondering if I should just replace part of the pipe with PVC and couplings or if I should replace as much as I can before the pipe goes to the foundation. I am an amateur handyman and I am not sure if this task is beyond my scope of abilities. Any advice would be much appreciated.

I have shoved a garden hose down the drain and run it full blast Before you start tearing up walls and slab why not open up what you already have.I have the same set up as you, house built in the 50's with galvanized pipes. Let me give you some options.
Rent a sewer machine and snake from the washer roof vent. Put out enough snake to reach the base and 18 feet more.
Now you can make your washer drain a closed system by connecting the washer hose to the stand pipe using a compression fitting or rubber coupling( See image) or you can do what I did and simply duct tape the hose to the stand pipe. That takes care of the backups. That way gets you past jackhammering up the floor and replacing pipes.
However, To install a utility sink the slab would have to come up and a new drain line cut into the main or the washer drain line which you'll be replacing at the same time. Your new sink can be revented back to the washer roof vent 6" over the top of the stand pipe or washer station. Just thought I'd give you a easy out. Good luck, Tomk