Normally, to rename a the field name of a dimension you can right click the field in the data window and select "Rename". To rename the field values, you can assign aliases (right click the field, select "Default Properties" -- "Aliases").
However, if you are using a data connection published to Tableau Server, you will not be able to rename the fields. In that case, you'll either
1. Create a local copy of the connection, rename fields as desired, then optionally republish to Tableau Server.
2. Create a copy of the fields, as you indicated above.
Hope that helps!