How to paint over a pre-existing problem?

Depending on the kind of paint used to paint over the marks, you may not be able to solve the issue without sanding that coat off. i.e. If the paint is peeling, new paint will not adhere properly (or if it does, will just peel off later.) If it was an oil-based paint, you can’t paint acrylic/water-based paint over the layer.

For the wall with permanent marker streaks, wash, sand, and prime the wall, then apply a finishing coat of the desired color. There are several primers available with extreme opacity, meant to cover up marks/etc. with just one coat. Ask your local hardware store for the options.