Pia, you really need to get bathroom paint, especially with a kiddie in the house!
The shop where you bought it will probably let you exchange it - I usually just smile sweetly and say 'I brought the wrong stuff apparently... / I am just a girl!?' a bit deemining I know, but it works for me!
There's not much point in painting the woodwork first, cos you might get splashes of paint on it and then have to do it all over again?
And please don't paint the tiles!