If you want the simplest solution, try using a bar of Ivory soap for your hand washables. Rub it on a section of the item, and also against any really dirty spots, then rub the cloth against other sections of the item, and do this until pretty much all of it has been "worked" with some of the soap. (note: you're only applying soap during the first rub, except for "stain removal"). Rinse well with tepid water.

My Mom used Ivory soap for both hand washables and dishwashing her whole life, and it seemed to work just fine, including on my Dad's cashmere sweaters. Me, I'm lazier , and prefer Woolite to the rubbing.