The renown (or friendship) system is a type of quest that can be done for specific NPCs once a day to earn favor from them. Doing this will unlock various rewards. You can view your progress in a specific renown quest by click the book icon on the top left of the screen.
Before V1.33 the renown system was a required part of the main quest line (with Sylvie Dawson and the dog being the only ones available at the time).
Each day the NPC you are befriending will have a task you must do, and is a random task from their list which you must find.
You can find the location he's at for the day by going to in-game channel /chat doggo or /chat renown, or by using this tracker. You can find a log of previous day locations here (although this has very little use as the order changes for anyone starting the quest).
Location: In the south district, outside the survivor's burrow (Where Eric Austin stands if his quest is available)
Objective: Walk the dog.
You'll be first prompted to go to the library, interact with the dog and then examine the table. Walk the dog to Tiscornia Park, and after the cutscene ends it will be at the memorial statue. Walk the dog to McKinley Park, and then find it behind the rubble on the right side. Kill 11 infected then interact with the dog.
Location: Pet Shop, Jellyfish Public Swimming pool
Objective: Wash, catch, dry, and scout for the dog
Enter the pet store (around the bowling alley). After interacting with the dog, find and collect 2shampoo bottles and a scrub, which can be found in the pet store. Scrubs and shampoos can be found near the pool (mirror in the left row of toilets, and under one the showers left of the pool itself), or in yellow janitor carts littered around the map, and in bathrooms.
Go to the Jellyfish Public Swimming pool. Next to the waterslide, you will find the place where you rinse the dog (a shower head connected to the wall and a bucket is present here).
To "interact with the world task," you go southeast of the slide alongside the edge of the pool, where there is a white shelf with towels and a beach ball under it. Click this.
Go back to the same spot by the slide to dry off the dog.
When it says to capture the runaway dog for the first time, go to the opposite side of the pool of the slide (long part) near the benches. The dog is here.
When scouting, go to the women's washroom (characterised by the pink tiled floors and the woman sign by the door).
The final runaway spot is back at the lounge (of the swimming pool, not the mall itself), where the sofas are located. Go to the far corner of that area.
Normal golf balls will not work! You must find "new" ones by search containers or by killing infecteds and search any green glowing loot bags (recommended). Next you must follow the dog to 3 locations and defend him while he digs.
Location: to the right of second bridge from spawn.
Objective: Bring dog to various water puddles. Protect dog.
Sometimes the final puddle doesn't generate enough of the "right" infecteds to count against your total - sometimes you'll get to 23 or 24/25 infected killed but it won't go to 25/25 no matter how many you kill in the area. If you leave the area and get far enough away the counter will reset to zero and it'll start spawning new, "right" infecteds to kill - you don't need to restart the quest from the beginning. You may need to do this a couple of times, though.
Location: Inside the Prison, in a hallway between the Basketball Court and the Sewing Room.
Objective: Lead Kelly Marsh to the safety of the campfire.
Find the dog in the prison; he should be in a hallway of some sort near the warden's office(1).
Follow the dog to the basketball courts. You will find Kelly Marsh, and will be required to defend her (kill zombies).
After killing the infected, you and the dog will escort Kelly Marsh to the campfire of the Refugee's Camp.
From here, you follow the dog to the restaurant near the prison and go to the bathrooms, where you will investigate what the dog is looking for - something to help Kelly Marsh with.
Bring what you had found to Kelly Marsh.
Location: Northwestern most part of Train Cars and Tracks
Objective: Defend Dog, search crates
Find the dog at the northwest edge of the train tracks and cars (1). Defend him from 15 infected, afterwards interact with him and then search the various crates directly near him until you find the correct crate he wants. He will then move down the train car and track line to (2). Again search the crates near him and after you find the correct one he'll move down the train line a bit more close to a backpack (3). Once more search the crates until you find the correct one. He will then move up to your right off the trail a tiny ways to a lamp post (4). Interact with him there, defend him from 20 zombies, then interact with him again.