In our organization, when you are looking at a report, the menu looks like this:
Can this be modified at all? For example, I'd like to bring the Save As and Export out of the dropdown, or even add a new button.
No, you cannot edit the menu other than by changing the user's reporting-related permissions to restrict specific operations.
The profile-level permission "Export Reports" can be removed to prevent users from exporting data.
The availability of options like Edit, Save, Save As, Delete, and Add to Dashboard will vary based on whether the user has Profile-level permission to perform such operations, as well as their ownership or access level of the specific Report in question. For example, if the user does not have "Create and Customize Reports" permission, they won't be able to edit any report.
See User Permissions for Sharing Reports and Dashboards for details on the available reporting permissions.