Necessities

Housing

Cage. See Cage and Housing Basics for recommendations.
Perches. Flat platform perches work best; thick rope perches made for large parrots are good for variety.
Bricks are important to have in the cage to keep nails in shape.
Bath pan. Any sufficiently large container will work.
Bowls for food, water, and grit. Unglazed terracotta bowls are recommended for food and grit to help keep beaks in shape; glazed ceramic or metal bowls are good for water. Specialized feed bowls for pets that prevent spillage (or even spoon rests as bowl) are another good idea.
Nest area like a cat house, a pet pillow, or even a large nest-like bowl.
Fake eggs if you’re getting a hen or aren’t sure of the bird’s gender.
HEPA air purifier. Pigeons are dusty, and a good air purifier is very highly recommended for your health.

Food. See Pigeon Diet Guide for recommendations.
Treats for training: a specific seed type that your pigeon likes but doesn’t have in their regular food mix. Safflower is a popular choice that most (not all!) pigeons will love.
Red grit mix. See Basic Grit Guide.
Cleaner such as Poop-Off or another enzyme cleaner.
Cage lining. Puppy pads are durable and easy to replace; newspapers are a cheap and abundant alternative; fleece blankets are less wasteful but need frequent washing.
Nesting material such as hay, straw, crinkle paper, or cardboard strips.
Wet wipes and paper towels for cleaning.