Identify the type of black paint that was used. If it is oil based then prime with an oil base primer. Kilz oil base works good.
Then, use a top quality flat paint for the first coat, mixed to the color you want. For white, have the paint store put a little extra titanium white in the paint for better coverage. Some manufactures call this "High Hide White". You can also have a little lamp black added to a white base. This is called "Knock Down White" and will cover better than a pure white paint.
Using a flat first coat will cover the black better than a satin or semi-gloss. After the flat dries you can paint with any sheen you desire and have better coverage.