Arbrex or other paint is not needed on any size of cut - I think has long since ceased to be recommended as it did not do anything to prevent infection. Ensure the cuts are as clean as possible - sharp tools and using loppers or saw for bigger branches, sloping cut to avoid water staying on it, and disinfecting tools regularly eg with rag and disinfectant. Do not prune too near to the adjacent branch or you will damage the branch collar and impair the healing (but not too far or you leave a stub which can more likely get infected) For the bigger branches cut above the place you want to prune first as the weight of the branch can cause the bark on the underside to tear back, and then cut the short remaining stub off at the correct point.
I do not know the best place on the internet about pruning but maybe the RHS has info and pictures which probably show things better
Simon