You’ve been given, through the efforts of Mr. Hodge and Mr. Martin, about all you’re going to find out about Francis Cochran. And you’ve been referred given the best reference for the history of his regiment. If you want a brief summary you can look at this site:
http://files.usgwarchives.net/ga/military/civilwar/rosters/57threg.txt
In regard to important battles, you should perhaps consider Champion Hill the most significant for the 57th Georgia regiment. But that battle was a part of the Vicksburg campaign, which included over a year of attempts to capture the city of Vicksburg, Mississippi. The fall of Vicksburg was a blow to the Confederacy that few events in the entire war could match. The campaign ended with a two-month siege of the city. The 57th Georgia regiment manned the trenches surrounding the city for the entire period of the siege without relief.
If you are trying to write a journal entry such as Francis Cochran might have made, you will have to try to put yourself in his shoes: a sick young man 600 miles from home surrounded by opposing land and naval forces which continually bombarded the city and cut off all supplies. People were living in holes dug into the clay of the bluff. The water and food were running out--famously, people started to eat their mules. It slowly became apparent that help would not be coming. Francis Cochran died of debility, basically he just didn't have enough strength to go on living. That is plenty to write about.
If you want to find out more about Vicksburg, here are some references:
• Ballard, Michael B. Vicksburg, The Campaign that Opened the Mississippi. Chapel Hill: University of North Carolina Press, 2004.
• Ballard, Michael B. Pemberton: A Biography. Jackson, MS, 1991
• Bearss, Edwin C. The Campaign for Vicksburg. Vol. 1, Vicksburg is the Key. Dayton, OH: Morningside House, 1985.
• Bearss, Edwin C. The Campaign for Vicksburg. Vol. 2, Grant Strikes a Fatal Blow. Dayton, OH: Morningside House, 1986.
• Bearss, Edwin C. The Campaign for Vicksburg. Vol. 3, Unvexed to the Sea. Dayton, OH: Morningside House, 1986.
• Carter, Samuel The Final Fortress: The Campaign for Vicksburg, 1862-1863, New York, NY: 1980
• Grabau, Warren E. Ninety-Eighty Days: A Geographer's View of the Vicksburg Campaign. Knoxville: University of Tennessee Press, 2000.
• Grant, Ulysses S. Personal Memoirs of U. S. Grant. 2 vols. Charles L. Webster & Company, 1885–86; reprint The Great Commanders, 1994
• Korn, Jerry War on the Mississippi: Grant's Vicksburg Campaign. Alexandria, VA: Time-Life Books, 1985.
• Miers, Earl Schenck The Web of Victory: Grant at Vicksburg. Reprint Baton Rouge, LA: LSU, 1984
• Smith, Timothy B. Champion Hill: Decisive Battle for Vicksburg. New York, NY: Savas-Beatie, 2006
• Walker, Peter F. Vicksburg: A People at War, 1860-1865. Chapel Hill, NC: 1960
• Winschel, Terrence J. Vicksburg: Fall of the Confederate Gibraltar. Abilene, TX: McWhiney Foundation Press, 1999.
George Walker