How does electronegativity change going down and across the periodic table?

Down a group, the electronegativity decreases. Across a period, it increases.

Down A Group
• As you move down a group, the number of energy levels increases. The valence e- cannot feel the pull of the nucleus as strongly.
• The nuclear shielding also increases, reducing the pull from the positive nucleus.
(Note: the nuclear charge [i.e. of p+] also increases down a group, rendering the nucleus more and more positive; however, the increased nuclear shielding compensates by decreasing the pull on the valence e-).

Across A Period
• The number of energy levels stays the same.
• The nuclear shielding also remains the same; thus, the valence e- has a weaker protection from the pull of the positive nucleus.
• The nuclear charge increases across a period, as the nucleus becomes more positive. This increases its pull on the valence e-, pulling it in closer.