How can I paint a white shiny wall board in my bathroom?

Hello Janice! You can redo your bathroom walls with a little bit of work. You will have to fill the nail holes. You can use drywall joint compound and do a bit of sanding once it's dry.

https://homeguides.sfgate.com/fill-finishing-nail-holes-before-painting-44715.html

Once you get your nail holes filled you will need to give your walls a light sanding to knock down the glossy paint finish. You could also use some Krud Kutter’s Prepaint Surface Preparation rather than sand. Follow the directions on the bottle. Once you wipe down the walls so that they are clean you can prime your walls. Once the primer is dry you can begin painting with latex wall paint. A satin finish paint will work well in a bathroom. You can also go with a semi gloss finish. I prefer the satin finish and have used it many times in bathrooms.

https://www.hunker.com/13416995/painting-over-semi-gloss-paint

https://www.wikihow.com/Paint-over-Gloss-Paint

Janice: Oh no! ...don't get new wall board! Merely fill in the holes with wood filler --- don't use too much, or you'll have to sand down. Then use a paint + primer.

Is it sheetrock or a laminate surface?

Take a good photo or wall sample to the paint department at your home improvement store and ask for their advice on paints and application tools!

I've painted this stuff before. You can fill nail holes with spackle, then make sure to prime it. I use Kilz, but if you skip this step or if you try to use a paint+primer instead, it will scratch right off. Once it has Kilz on it, you can paint like normal.

Sounds like bath panels over wallboard. Fill the holes with no shrink drywall patching compound. Let dry and sand smooth. Use a green kitchen scrubby to remove the gloss. Don't be too aggressive so not to harm the coating. Prime with Kilz primer. Paint with a good bathroom paint. Has mildewcides to slow down mold and is moisture resistant.

Unless you find something to plug/smooth out the holes, I would just get new wall board.