When creating a new story or bug you can add a dependency or relation to other stories. Is there a way run a report to show all these stories with these relation stories?
A first option would be to build a JQL filter to show all issues with linked issues. In your search results, you can display the Linked issues column, which will display all linked issues.
Unfortunately, this functionality is not available out of the box, so you will need an add-on for this. For a few options (Scriptrunner, Search Linked Issues for JIRA, JQL Tricks), see this related Community question.
Recently, Portfolio for JIRA has added a dependencies report, which is probably coming closest to what you describe. You would normally not set up Portfolio just for the Dependencies Report alone, but if you do, it's a nice additional asset.