technically speaking, your query is just fine, and the process to show each itinerary only once, combining the countries underneath, should actually be done by your application layer language (php or whatever)
it is also possible to do it with mysql, but only when there is a single one-to-many relationship involved, and when the many data is limited to one or two columns of data…
… which is the case here, so you’re in luck 
SELECT itineraries.ItineraryID
, itineraries.UserID
, itineraries.LodgeID
, itineraries.Itinerary
, itineraries.Summary
, itineraries.Day_1
, itineraries.Info_1
, itineraries.Night_1
, itineraries.Day_2
, itineraries.Info_2
, itineraries.Night_2
, itineraries.Day_3
, itineraries.Info_3
, itineraries.Night_3
, itineraries.Day_4
, itineraries.Info_4
, itineraries.Night_4
, itineraries.Day_5
, itineraries.Info_5
, itineraries.Night_5
, itineraries.Day_6
, itineraries.Info_6
, itineraries.Night_6
, itineraries.Day_7
, itineraries.Info_7
, itineraries.Night_7
, itineraries.Day_8
, itineraries.Info_8
, itineraries.Night_8
, itineraries.Arrive
, itineraries.Depart
, itineraries.Duration
, itineraries.Customise
, itineraries.Country
, itineraries.Airfare
, itineraries.Supplement
, itineraries.Persons
, itineraries.Children
, itineraries.Accommodation
, itineraries.Game_Parks
, itineraries.Currency
, itineraries.Price
, itineraries.Price_Supplement
, itineraries.Details
, itineraries.Valid_From
, itineraries.Valid_To
, itineraries.Publish
, itineraries.Publish_GSG
, itineraries.Publish_SB
, itineraries.Publish_SL
, itineraries.Image1_title
, itineraries.Image2_title
, itineraries.Image3_title
, itineraries.Image4_title
, itineraries.Image1
, itineraries.Image2
, itineraries.Image3
, itineraries.Image4
, itineraries.Itinerary_PDF
, [B][COLOR="#0000FF"]GROUP_CONCAT(activities.activity) AS countries[/COLOR][/B]
FROM itineraries
INNER
JOIN ItineraryActivities
ON ItineraryActivities.ItineraryID = itineraries.ItineraryID
INNER
JOIN activities
ON activities.ActivityID = ItineraryActivities.ActivityID
AND activities.Category = 'Countries Visited'
WHERE itineraries.Publish_GSG = 'Yes'
[B][COLOR="#0000FF"]GROUP
BY itineraries.ItineraryID[/COLOR][/B]
tested okay – see itinerary 247