Click "Edit" beside the entry you would like to edit at your trip journal. At the right bottom corner you have a link "Delete this post". After clicking this link you will be asked if your are sure to delete this post. Choosing "Yes" deletes you post.

To delete your whole trip journal choose "Trip Settings" and scroll down to the bottom of the page. There is a link "Delete this Trip Journal". Choosing "Yes" wil delete you trip journal. This step cannot be undone!