Flat slab design entails analysing and designing an orthogonal grid of flexural elements, supported by columns at the intersections of these elements.
The slab forms these flexural elements, with the element width equal to the sum of half the support spacings on each side of the support column. The centre portion of the flexural element (the column strip) takes a greater proportion of the total bending moment, relative to its width. Your design code should give you the proportions of the bending moment split, as well as rules for checking punching shear etc.
To design a flat slab, you first need to figure out the thickness of the slab for deflection control. Next based on the layout of your framing and load distribution, you can either use dirrect method or equivalent frame method to design the slab. Follow ACI 318 chapter-9 for slab thickness requirement and chapter-13 for design method.
