I missed my lesson on periodicity so had to teach myself, and have always forgotten to ask my teacher to explain to me why these trends are seen, which, unfortunately, the textbooks don't.
Density: the density of an element tends to increase across a period up to group 3, and then starts to decrease and become very low at group 5 or so. Why is this?
Melting and boiling point: they tend to rise up to about group 3 or 4, before falling rapidly after that. I would suggest that this is because group 4 elements tend to have giant covalent structures, so also a higher boiling point, whereas on the right hand side of a period they are simple molecular structures, so have a low boiling point. Am I correct?
Atomic size: atomic size always decreases along a period from left to right. My suggestion is that this is because there are more electrons and more protons as the group increases, so there becomes a stronger attraction between them, pulling the electrons closer to the nucleus. Am I right?
So the only one I have no idea about really is density, and the other two I just need clarification on.