Pour the paint into the base of a paint tray, then push a roller cover onto a roller, and apply paint to the roller cover by moving it backwards and forwards over the paint tray's bed. Allow an ample amount of paint to soak into the roller cover, but not to the extent where it starts to drip.
Roll the paint backwards and forwards onto roughly a 3-foot square section of the wall or ceiling. Load up the roller cover with more paint if you need to, and roll backwards and forwards over the painted section from a perpendicular angle to the first coat. This process helps cover all nooks and crannies of the area, especially if the surface is textured.
Run the tip of a 2-inch (or wider) bristle brush over the entire painted area, roughly at a 45 degree angle to the surface, using a minimal amount of pressure. The brush marks will disappear as the paint self-levels and dries.
Continue onto the next 3-foot square wall/ceiling section following the same process, overlapping the sections by roughly an inch. Carry on in like fashion until the entire area needing to be painted is completed.