"Seeing the elephant" is an American neologism that was probably
a revision of the old English idiom "see the lions" (referring
literally to seeing lions kept in the Tower of London, but also
metaphorically to seeing something new and exciting).
The American version refers to gaining experience of the world at a significant cost. It was used variously to refer to seeing battle (especially during the Mexican-American or Civil War), experiencing the 1849 Gold-Rush, and the experience of westward travel by pioneers over the "Westward Expansion Trails". It eventually also became a euphemism for visiting brothels and saloons. It was most commonly used during the 19th century but still gets some use in the military context referring to seeing battle - something that sounds exciting at first but quickly becomes something else when faced with the reality of the horrors of war.
At least two books about the Civil War use the phrase in their titles:
"Seeing the Elephant": Raw Recruits at the Battle of Shiloh
Seeing the Elephant: A Story of the Civil War