I checked in the location as the red arrow on the map pointed out, and by the time I got there, I saw the dog by the wooden fence,but then he left and suddenly vanished, though the yellow arrow indicated his initial spot. What am I supposed to do now?

Hi

This is a common bug that may happen with the quests that involves a NPC moving that you have to follow.

I suggest that you relog or try an other server, as crowded server seems to increase the chance this problem happend.

If you can't get it to work, and if it's the main quest you're talking about, you can still trigger the next quest by walking to the house with a pool, north east of the map's entrance.