One that gives a good perspective on the war and its barbarization is Willy Peter Reese's A Stranger to Myself. Unlike many other personal accounts from Wehrmacht soldiers, this one is posthumous and thus not sanitized for a postwar audience. Reese was a somewhat typical soldier in the sense that there was not much about him that was unusual- he was no National Socialist, but was influenced by them. He was no frothing antisemite racist, but he nevertheless participated in acts of wanton brutality against the civilian population of the USSR.

For the wider social background of the army, be sure to check out the secondary source Hitler's Army: Soldiers, Nazis and War in the Third Reich by Omer Bartov, which has a lot of statistical information such as NSDAP membership among the officer corps, that helps contextualize these memoirs.

Soldat by Siegfried Knapp is a personal account of his time in the Wehrmacht, from the invasion of Poland through Soviet capture in '45. You have to look at what he says through a bit of a filter; sometimes he comes off sounding like "National Socialism was bad, but nobody knew!" and downplays/plays apologist for certain things, but the nuts and bolts of living in Hitler's war machine are spot-on.

Great book, not a difficult read at all.

Blood Red Snow - The memoirs of a German soldier on the Eastern Front. By Gunter K. Koschorrek.

A brutal and detailed account of the experiences of a German heavy machinegunner, fresh out of basic training in 1942, through to the end of the war.

A truly fascinating read.

I recommend "Panzer Commander: the memoirs of colonel Hans Von Luck". It's the perspective of a commander, no a private soldier, but he was on both the eastern and western fronts so his memoir has a lot of insight into the war from a German perspective.

You could try Soldaten which contains covertly recorded conversations between German POWs.

Repulsive, revolting, repugnant, and utterly intriguing.

Otto Carius' Tigers in the Mud was an interesting account, but like all memoirs, take what he says with a grain of salt.

I cannot remember the name of it, but there is a amazing memoir written by a solider at the Battle of Stalingrad that captures the horror of that event. Ill get the name of it, it's at home somewhere. Also not really what you are looking for, as it deals with the Hitler Youth, but A Child of Hitler by Alfons Heck is a very good read. Gives great insight into how the Hitler Youth was run and how these children were indoctrinated.

Black Edelweiss: A Memoir of Combat and Conscience by a Soldier of the Waffen-SS by Johann Voss is an interesting and insightful read, because it details his combat experience in the Arctic circle as well his pre-and post-war thoughts and experiences. Will give you an insight into the social and political views of what we can assume is a (somewhat) typical educated young person during the Nazi regime.

The Forgotten Soldier by Guy Sajer

Sajer was a German living in the Alsace region of France who was drafted by the German Army in WWII. He ended up in the elite Grossdeutschland division and served in several campaigns in Russia. There is some controversy as to the accuracy of this book due to conflict between the location of the division and some of the battles but it is still a fascinating read.

I've read the other books recommended by other users, and give them all a collective shrug...one not mentioned by others that would be worth your time is Bidermann, G. H. (2000). In Deadly Combat: A German soldier’s memoir of the Eastern Front (D. S. Zumbro, Trans.). University Press of Kansas.