GEDCOM
In theory exporting your GEDCOM file from Family Historian can be done as simply as sending the .GED file located within your Project folder. However, in many cases it is much easier to decide exactly what you want to send to the contact, and remove excess information before you start. Both of these processes can be done using File > Import/Export > Export > GEDCOM File and / or the Split Tree Helper.
There are many things to consider when exporting data to another user, not least the privacy of the data that you already have on hold. Both tools mentioned above can help by removing living people using a query, and exclude record type data that another program may not understand. To remove specific events, or local notes, use the Split Tree Helper.
Other products may not understand Multimedia Records, Repository Records, Place Records, and sometimes even Source Records. This is especially true of online trees on FamilySearch, Ancestry, FindMyPast etc, which are predominantly designed to share very basic information and not the levels of information which are typically stored in Family Historian. The Export Gedcom File Plugin is useful for exporting in the specific GEDCOM dialect required for other genealogy products.
Diagrams
Diagrams can be exported in a variety of formats by simply using the Save Diagram As dialogue on the File or Diagram menu. This includes PNG format, and a selection of standard formats including EMF, bitmap, tiff and Photoshop format. However many users prefer to use PDF as this is globally recognized and easy for many users to read. See how to create Single Page PDF of a Diagram either using later versions of Family Historian or a third party program.