What were some battles that led up to the US being able to get within bombing range of Japan.
In essence, every battle after Pearl Harbor (or at least after the Philippines had been lost) can be seen as a battle that helped to get the US within bomber range of Japan. Having been pushed out of the Philippines and Guam, the US was trying first to prevent the Japanese from expanding any further and then to push them back towards the home islands.
There were, however, a few battles that were immediately beneficial for allowing the US to be able to bomb Japan. The first of these was the Battle of Saipan. The US invaded the island of Saipan and fought a very bloody battle to defeat the Japanese defenders. With Saipan taken, the US was easily able to take the neighboring island of Tinian, where it built huge runways that allowed B-29s to take off and bomb Japan. During the Battle of Saipan, there was a naval aviation battle called the Battle of the Philippine Sea that also helped to secure Saipan and Tinian. Finally, we can say that the Battle of Iwo Jima helped as well. It gave American planes from the Marianas a place to land in emergencies and was eventually used as a launching point for some bombing raids.