In Pearl Creek, at the intersection of Cook St and Lansdale Ave, find a building with a missing person poster next to the main entrance:

Ariadne Johnson
DOB: January 6th, 1996
Missing: April 11th
Height: 4' 2"
Eyes: Brown
Race: Caucasian
Age: 8
Sex: Female
Weight: 48 lbs
Hair: Dark Brown
Ariadne was loast seen on April 11 around 2:15pm while walking home from the Hillside Middle School. Ariadne suffers from severe autism and is prone to wandering away from home. She follows a fixed route home from her bus stop, marked by a series of colored ribbons placed by her mother. Ariadne has a strong aversion to the color blue and has been known to alter her route to avoid the color.
Anyone having information should contact the silent Hill Police Department's missing persons hotline.

Reading this poster will start the sidequest Ribbons and give you your objective, to find the missing girl.

Follow Lansdale ave north, and when you hit its intersection with Brite st, near the phone booth, there is a lamppost with a yellow ribbon on it. On the ground, there is a lunchbox with a mystery note from the girl's mother:

Mommie has made a new path for you to follow home. Remember the rules...
Yellow Ribbon - Turn Left
Red Ribbon - Turn Right
I know you sometimes get confused, so just be a big girl and look for mommy's ribbons and you'll be fine. Whatever you do, DON'T BREAK THE RULES! Mommy will be waiting for you at home.
Hugs and kisses,

Follow the route of the ribbons. When you see a yellow ribbon you need to turn left, and a red ribbon means to turn right. Blue ribbon means to continue straight. If you get lost, refer to the maps below which will show you the location of your final destination (you don't have to follow ribbons, you can go straight there).

SIlent Hill Downpour Map for Side Quest RIbbons

1: Missing Persons Poster - Quest Start
2: Lunchbox and Yellow Ribbon

From where you found the lunchbox and note, go a bit north. There is a barricade there, slip through the gap on the west.  When you reach the road you should see a red ribbon on there, so turn right. Cross through Logan's Park. If that gate is locked for you, either continue with the main story, or use the underground shortcuts which you should have access to after completing the Homeless quest.

You will then have to go through the alley near the movie theater then turn north.

Silent Hill Downpour Ribbons map

3: Key to girl's apartment

Eventually you reach pleasant river, and the ribbons lead you to the docks. On the docks, red ribbons will lead you to a specific dock. There is no sign of the girl, but you do find a key on the edge of the dock. Your task now is to find the girl's home. The clue is in the key, so examine it in your inventory:

Found on a bracelet. Labeled "Lamon Street".

Silent Hill: Downpour Ribbons Sidequest Girl's Apartment

4: Girl's Apartment

The key leads you to Chastain Heights, and find an unlocked house on Lamon St., north of the movie theater. Climb the stairs and use key to open the door on the second floor. Explore the apartment. Find the bedroom. On the bed, there is a note from the mom, confessing she placed the ribbons that way to lead her daughter to drown on the docks. The note reads:

I've finally done it. I can't take it anymore. God knows I've tried to give her a happy, normal life, but I'm tired and worn out and I just can't do it another minute longer. People will judge me and call me a monster and wonder why, but until they've lives with a child like her, they can't know, CAN'T POSSIBLY KNOW, how hard it is. Every single day, every moment. She will NEVER be normal. She will NEVER grow up and lead a happy life. I've been living a lie, and it's time to wake up from the nightmare.
I've given all I can possibly give. It's time for me to start living my life for ME. Don't try to find me. You'll never see me again. My new life starts TODAY
God forgive me,

This completes the Ribbons Side Quest, and unlocks a trophy/achievement:

Long Walk, Short Pier
Completed the "Ribbons" side quest

After completing the Ribbons quest, another door will unlock in the apartment, near where you entered. This is the little girl's room, and there are two first aid kits inside as your reward for completing this quest.