This is your first Quest from Mayor Barley and is given on the first occasion you speak to him after the game starts proper on Spring day 2.
When you talk to him he asks how you are. Any response leads to the suggestion that you should introduce yourself around town.
'You can use this checklist to keep track of the folk you've met and those you haven't.'
You receive the Villager Checklist which is accessible in your knapsack. Names are crossed off as you meet people. If you talk to the Mayor again during the Quest, he will tell you how many people you have already met.
This is fairly straightforward and can be accomplished as you make your way around on the first game day.
For help see the Characters page.
When you complete the Quest you will record the mark 'Social Creature'
If you go back and talk to the Mayor now, he will give you the Village Primer. This book has profiles of all the villagers and helps you keep track of relationships.