Atlanta Hardwoods went out of business. Sad news, they were a great source.
Good prices on BB ply and MDF at Rugby Architectural Building Supplies. They have multiple thicknesses. Call ahead for availability, they have a will-call window from what I understand.
Specialty woods and veneers in midtown called Carlton’s:
Highland woodworking is a good source:
Rockler has multiple locations in the city:
(and maybe more)
Woodcraft has two locations around town:
(Woodcraft of Atlanta, and Woodcraft of West Atlanta. The others are not the same.)
Professional Plastics has good deals on acrylic sheet, especially if you can pick up locally:
Uline in Gwinett county:
Sam Flax art supplies in midtown has many materials:
IKEA in Midtown:
The following places will do in a pinch but are not recommended due to price/other considerations.
Hobby Lobby & Michael: Expensive for what you get, but reliable source of hobby wood and other almost ready-made things like frames and plaques.
Big Box hardware: very limited thin stock
Ace Hardware locations: Some thin stock, but unreliable supply.