The "big three" are Isaac Asimov, Robert A. Heinlein, and Arthur C. Clarke. It reminds me of a book, whose author and title escape me, where all the famous science fiction writers of the time are summoned to help in a global emergency. Asimov, Clarke, and Heinlein are depicted as the elders of the group - impatient with the younger members.
These were not the first science fiction writers, so they did not define the genre. Great writers like H. G. Wells, Jules Verne, and Edgar Rice Burroughs had already seen the potential for science to drive a story. But what the big three did was expand the definitions of the genre - particularly the style called "hard science fiction," where it became important to stick to the laws of Physics as much as possible. Of course, these writers had a generation of knowledge the previous authors did not. But one comment you will hear a lot is that their novels have been shown to have predicted the future. Wells' stories are wonderful, but when you see pictures of the space station, you can't help but recall Clarke's 2001: A Space Odyssey. Asimov's Three Laws of Robotics seem ever more pertinent when we hear that the U.S. Armed Forces are considering using robots on battlegrounds - to recover injured soldiers. These are just the crudest examples.
It could even be argued that it was these later writers that got the genre taken seriously as literature. Through this huge body of work, we see the novels of Wells and Verne in a new light. I would not take this argument too far though.
And so I would argue that it is these writers that influenced science fiction the most. Science fiction readers, particularly of my generation, would have read their works extensively, and this would have been coupled with an interest in cutting edge science.
For me, Heinlein was best. His novels span several types - juvenile, hard science fiction, and more philosophical books like Time Enough for Love and Stranger in a Strange Land. In one of my favorite books, Tunnel in the Sky, a group of children is teleported to a planet for a three day survival course. When the teleportation device is compromised, they are stranded and must really work to survive. Another classic is Friday, a fast paced thriller starring a female operative. This was my first Heinlein, given to me by an old friend.