What is actually happening
A razor bump is a hair that has curled back into the skin, and the redness around it is your immune system reacting to it as a foreign object. That process starts hours after the shave, which is why the bumps show up the next day.
A sharper blade helps at the margin. What helps more is reducing the inflammation that is already there before the hair grows back through.
The order that matters
Lather properly and give it thirty seconds to soften the hair. Shave with the grain, not against it, even if the result is slightly less close. Then get something anti-inflammatory and alcohol-free onto the skin immediately, while the follicles are still open.