How to stop old wall paint from peeling underneath when trying to paint over it?

Start by selecting appropriate colours. They if you are painting light on dark you will need a white paint to use as a mist coat (50 -50 paint and water if using water based paints) if going darking the same process using a darker paint to drop the colour as a mist coat.

Start by filling cracks and holes then lightly rubbing down walls with a fine sandpaper to make a key for the next coats to attach to.

Lay floor covering

Apply the mist coat to the walls and allow to fully dry

Check all walls again for any dirt specks sand lightly once more

Start by cutting in around the room( use a 2 inch quality brush to get to straight edges i.e. adjacent to ceiling . if required use a masking tape if you are not confident brushing straight edges

Using a medium lambswool roller roller the way using the "W" pattern

Allow to fully dry and if required apply 2nd coat