I've used a magic eraser, but it can take off the paint. I also repaint mine every 4 years or so, that helps. Yes to vacuum, dust and hands and knees with a sponge. When I lived in a humid climate they were harder to keep clean. Now that I live in Colorado, it's a bit easier.

I do mine occasionally, maybe once a year, with a bucket of soapy water and a rag, on my knees. I think you get better friction with a rag than with a sponge. An old towel works well. I don’t try to do the whole house at once, maybe one or two rooms a week, as long as it gets done eventually. That reminds me, spring cleaning.

Major sure your baseboards have an enamel or glossy paint to enable easiest cleaning.

High gloss hides a number of flaws and depending on the color it also hides dirt. The only ways to fix it is to clean one room at a time or whole house in a weekend and keep it on the cleaning task schedule after. Either or you still have to get it done and then maintain. Spring cleaning - have you cleaned the part of the wall behind your headboard since the bed was first set up? And vacuumed under the bed fully? Those two tasks I hate but have to get them done anyway - the mister has to move the bed for me though, too heavy to do both the moving and the cleaning those spots before remaking the bed with clean sheets all by myself.