@Birgit - Welcome to the forums. You could try the Learn JIRA Basics course from the training site. Excellent for new users with little prior knowledge, and just over an hour to complete.
I have been through all the basics, but Atlassian does not do a good job bringing the pieces together, which makes it very hard to actually implement Jira.
Are there tutorials that actually talk about setting up Jira, all the pieces to consider and company wide implementation and training?
@Birgit - apologies for taking so long to reply. The best resource to cover all your requirements would be the various learning paths within Atlassian University.
Choose the user type (JIRA Admin, Project Admin, User) and follow the required learning. There will be free learning as part of all these paths, as well as a certified completion should your Admins require it.
For the three areas you mention, setting up JIRA is a JIRA Admin task, company wide implementation would largely fall to a group of passionate Project Admins, and users receive training from both the above.
As a fallback, the documentation is a good place to start, as are these forums for all your questions.
