In Sonic's story, he goes to Flame Core to escape Crisis City and retrieve a Chaos Emerald. He starts off outside the volcano then travels inside it. Sonic then fights Iblis Phase 2. After that he does a "Cross-Chaos Control" with Shadow to escape to the present.
In Shadow's story, he goes to Flame Core to escape Crisis City and find a Chaos Emerald. His version of the stage is slightly harder than Sonic's. He starts off outside the volcano, then goes inside. Shadow fights Iblis and Mephiles-type monsters. Afterwards he fights Iblis Phase 2. Then he "Cross-Chaos Controls" with Sonic, but then decides to stay behind and fight Mephiles. Shadow is later sent back to the present because he (along with E-123 Omega) follows Mephiles via a time warp.
In the final level in Silver's story, he goes to Flame Core by using "Cross-Chaos Control" with Sonic, rushing to Iblis. He starts off outside the volcano and goes inside it. Silver fights Iblis and Mephiles monsters. At the end of the stage he destroys a giant ball of flame, which is part of Iblis. Afterwards he goes on outside and fights Iblis Phase 3.
In Sonic's Flame Core in Very Hard Mode, he actually starts out inside the volcano. He encounters Knuckles, like the original stage. However, when Sonic reaches the end of the volcano, there is a warp instead of a Goal Ring. This warp takes Sonic back to the beginning of the outside section of the stage, where he must get through more traps than the usual stage and reach the end of the outside section, where the Goal Ring resides.
Shadow starts out inside the volcano as well, but at the end of the stage. After making a comment about going upward, Shadow comes out of the final shaft that he usually runs down to finish the stage. Rouge then takes over and she needs to find keys to move on.
There is a glitch in Sonic's Flame Core stage where you can finish his stage as Knuckles, by climbing a cornered wall and gliding, however to see his stage finish screen the player must hit the last checkpoint.