As you move along a period in the periodic table, elements gain protons, increasing the atoms nuclear charge. This larger positive charge has a stronger attraction to the negative electrons in the outer shells, holding them closer to the nucleus. This reduces the atomic radius of the element, explaining the trend we see across a period. As you go down a group, the atomic radius increases as there's a new shell of electrons.
There is no strict trend that all periods or groups follow in terms of melting point. The melting point of an element depends on the type of bonding it forms. The melting point of metals (Na → Al) increases with valency, as its bonding strength depends on the volume of delocalised electrons present. Elements which form large strong covalent structures (Si) have the largest melting point. Molecular elements (P → Ar) are only held together by weak van der Waals forces, so have the lowest melting point.
The 1st Ionisation Energy of an element is the amount of energy required to remove the first valence electron, forming the X⁺ cation. As the atomic radius decreases, the nucleus has a tighter hold on its electrons, so more energy is required to remove them. Therefore, the 1st Ionisation Energy increases across a period and decreases down a group. However, there are exceptions to this general trend, due to the type of sub-level the electrons are being removed from.
The amount of energy required to remove the second electron from a singly charged cation is known as the 2nd Ionisation Energy. It follows a similar trend to 1st Ionisation Energy, depending on atomic radius and electron configuration. The 2nd Ionisation Energy is a lot bigger than the 1st Ionisation Energy for the same element, as there's reduced electron repulsion.
The electronegativity (χ) of an element describes how readily it attracts a shared pair of electrons towards itself. The electronegativity of an element increases as you move from the bottom left to the top right corner of the periodic table. There are many mathematical definitions of electronegativity, the most commonly used is The Pauling Scale.