Walk in a vain, self-important way

One word substitute is Strut.

Jog: run at a steady gentle pace, especially on a regular basis as a form of physical exercise.
Trek: a long arduous journey, especially one made on foot.
Trudge: walk slowly and with heavy steps, typically because of exhaustion or harsh conditions.
Strut: walk with a stiff, erect, and apparently arrogant or conceited gait.