Always worth a look at https://confluence.atlassian.com/jira/migrating-jira-to-another-server-151126305.html
These are the steps I've used.
install JIRA on the new server as normal
Schedule down time
Disable logons except for admins
Create back of data using built in function (I suggest using zip file option)
Import backup on new server using built in function (this can take a while)
Move folder with attachments to new server
Verify everything by logging on using the localhost URL
If everything is OK update DNS to point to new server
Reenable logons for users
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