Remember that YOU are your pet's best advocate. You have the best information about how your pet behaves in a home, what toys/treats it likes, etc., so you are best equipped to find the right home for it. Before turning to your local shelter for rehoming, try speaking with family members or friends about taking your pet in first, and/or contact local rescue groups for resources/placement.
If you are surrendering your pet due to behavioral issues, please see these behavior archives for help dealing with common behavioral challenges:
Dogs: https://www.aspca.org/pet-care/dog-care/common-dog-behavior-issues
Cats: https://www.aspca.org/pet-care/cat-care/common-cat-behavior-issues
If you are experiencing financial strain, there are low-cost resources for food, housing, medical care, and more below: