What You Need

A sink or a large bowl
Cold water
Mild soap (I use dish soap. Not the fancy kind.)
A white towel you don't love
A place to hang or lay flat
What I Messed Up So You Don't Have To
The first old dress I washed went straight into the machine. Delicate cycle. Cold water. I thought I was being careful.
The dress came out two sizes smaller. The sleeves looked like they belonged on a doll. I actually cried a little.
The second time, I used hot water because I wanted to "really clean it." Bad idea. The color ran everywhere. A yellow dress turned pinkish gray. The trim was supposed to be white. It wasn't anymore.
Now I stick to cold water and my hands. That's it.
How I Finally Got It Right

Step one: Fill the sink with cold water
I use the coldest setting. Not room temperature. Cold. Old cotton fibers relax in cold. They shrink in anything warmer.
Add a drop of dish soap. Just one. Too much soap leaves residue and you'll need to rinse forever.
Step two: Submerge and wait
Push the dress down until it's fully wet. Walk away.
I leave mine for fifteen minutes. Not more. I set a timer because I forget. Left one dress for an hour once. The water turned brownish. Not from dirt. From the fabric aging faster than it should.
Fifteen minutes is enough.
Step three: Rinse with cold water
Empty the sink. Refill with clean cold water. Swish the dress around. Repeat until no bubbles.
I usually do this three times. Sometimes four if I used too much soap.
Step four: Press, don't wring
This is the part I messed up most. I used to twist the dress to get water out. Twisting breaks old fibers. You won't see it right away. You'll see it six months later when a sleeve tears at the seam.
Now I lay the dress flat on a towel. Roll the towel up like a sleeping bag. Press down with my knees. Unroll. The dress is damp but not dripping.
Step five: Dry flat or on a padded hanger
I lay mine flat on another towel. Never in the sun. Sun fades old colors. Never in the dryer. The dryer will kill a 60-year-old dress in twelve minutes.
If I use a hanger, I pad it with a small towel so the shoulders don't stretch.
One time I washed a 1940s housedress and a button fell off in the sink. I fished it out and sewed it back on. That button is slightly looser than the others. I notice it every time. No one else has ever said anything.
Try this with something old but not precious. A scarf from a thrift store. A linens tablecloth. See how it goes. Put a photo in the comments. I want to see what you saved.