You’ll trigger this quest as soon as you arrive in Stardew Valley. All you need to do to complete it is speak to the NPCs in town. You’ll start with two of them from watching the intro; now you need to find twenty-six more. You can find a listing of the NPCs, met and otherwise, under the Social tab of the main menu.

Below is a listing of where you’re most likely to find each of the NPCs. The first entry is during the day; the second is later in the day, or at night. NPCs move around often and will change their schedules depending on the weather, so you’re not guaranteed to find them in these places. You will, however, find them in these locations if you sit around long enough. Note the outfits of some of the NPCs - if they change, the NPC is probably at their job, not at home.

(Guide in progress. If you have better locations for these NPCs, please let me know in the comments and I'll update this list accordingly.)

The ordering of the NPCs under the Social tab changes depending on your relationship with them, so it's going to jump all over the place as you play. For sanity's sake, everybody’s in alphabetical order below.
  • Abigail - Pierre’s General Store (centre of Pelican Town)
  • Alex - 1 River Road (just east of the centre of Pelican Town - he often likes to lock himself in his room; try earlier in the day)
  • Caroline - Pierre’s General Store (centre of Pelican Town)
  • Clint - Blacksmith (southeast of Pelican Town), the Saloon (centre of Pelican Town)
  • Demetrius - Carpenter’s Shop (north of Pelican Town)
  • Elliott - Elliott’s Cabin (on the beach south of Pelican Town - you can’t enter his cabin most of the time so you need to wait around until he appears; occasionally he shows up in Pierre’s General Store)
  • Emily - 2 Willow Lane (south street of Pelican Town), Saloon (centre of Pelican Town)
  • Evelyn - 1 River Road (just east of the centre of Pelican Town)
  • George - 1 River Road (just east of the centre of Pelican Town)
  • Gus - The Saloon (centre of Pelican Town)
  • Haley - 2 Willow Lane (south street of Pelican Town)
  • Harvey - Harvey’s Clinic (centre of Pelican Town)
  • Jas - Stardew Valley Museum & Library (southeast of Pelican Town), Marnie’s Ranch (south of your farm)
  • Jodi - 1 Willow Lane (south street of Pelican Town)
  • Leah - Leah’s Cottage (south of your farm - you can’t enter, so unless you find her wandering around the streets you’ll need to wait out front until she comes home)
  • Lewis - Met automatically
  • Linus - Tent (north of Pelican Town, behind the Carpenter’s Shop)
  • Marnie - Marnie’s Ranch (south of your farm)
  • Maru - Harvey’s Clinic (centre of Pelican Town) Carpenter’s Shop (north of Pelican Town)
  • Pam - Trailer (just east of the centre of Pelican Town), the Saloon (centre of Pelican Town)
  • Penny - Stardew Valley Museum & Library (southeast of Pelican Town)
  • Pierre - Pierre’s General Store (centre of Pelican Town)
  • Robin - Met automatically
  • Sam - 1 Willow Lane (south street of Pelican Town - he likes to sit in his room for a looooong time, so you might have to loiter in his house a while in order to catch him on the move)
  • Sebastian - Carpenter’s Shop (north of Pelican Town - he seems to lock himself in the house most of the day, so you’ll have to wait for him to leave his home, around 7 pm - look to the east to find him smoking by the water)
  • Shane -The Saloon (centre of Pelican Town)
  • Vincent - Stardew Valley Museum & Library (southeast of Pelican Town), 1 Willow Lane (south street of Pelican Town)
  • Willy - Fish Shop (south of Pelican Town - triggered by getting his letter on the second day)
And what do you get out of finding all those NPCs? Nothing!  No prize!That’s what! All you really get is the quest How To Win Friends, which is substantially easier to complete.