The two forces behind every trend
Only two things drive all of these trends. First, nuclear charge: more protons pull the outer electrons in harder. Second, shielding: each extra electron shell pushes the outer electrons further out and blocks some of that pull.
Across a period, protons increase but the number of shells does not, so the pull wins. Down a group, shells increase, so shielding wins. Every trend below follows from those two sentences.
Atomic radius
Decreases across a period (stronger pull, same number of shells) and increases down a group (more shells). Sodium is much larger than chlorine; caesium is much larger than sodium.
Ionisation energy and electronegativity
Ionisation energy is the energy needed to remove an outer electron. It increases across a period and decreases down a group — the closer and less shielded the electron, the harder it is to remove.
Electronegativity is how strongly an atom pulls shared electrons in a bond. Same pattern: increases across, decreases down. Fluorine is the most electronegative element; the alkali metals at the bottom left are the least.
Metallic character and reactivity
Metallic character is the opposite of electronegativity: it decreases across a period and increases down a group. Metals lose electrons, so reactivity for metals increases down the group (potassium reacts far more violently than lithium). Non-metals gain electrons, so their reactivity increases up the group (fluorine is more reactive than iodine).
Answer template for trend questions
State the trend, then justify it with nuclear charge and shielding, then give the example from the question. 'Atomic radius decreases across period 3 because nuclear charge increases while the number of shells stays the same, so the outer electrons are pulled closer; hence Na is larger than Cl.' That structure earns full marks almost every time.