When Grant enters West Point he wants to correct the record to show that his true full name is Ulysses H. Grant, not Ulysses S. Grant. That is not true. Grant's birth name was Hiram Ulysses Grant, but he hated "Hiram" and was called "Lyss" or "Ulyss" by his wife and family. He got the middle name of Simpson at West Point because that was his mother's maiden name. The authorities at West Point noted that he had no middle name in his paperwork and someone who knew the family, and going with the tradition of the time (and even now), assumed that his middle name must be his mother's surname, hence Ulysses Simpson Grant.
During an early attack by Union forces, a Sergeant is shown leading an infantry unit. The Sergeants stripes are red, stripes for infantry NCOs were blue, red stripes were worn by artillery NCOs.
The siege of Fort Donelson was fought in the winter (February 1862) and conditions were cold and miserable, not sunny and apparently warm as shown.
The Battle of Belmont was fought in MIssouri, not Kentucky.
Grant sits on his horse early in the morning at an intersection and directs his army to march south, further into Virginia. As the troops march right to left across the screen, the rising sun is behind them in the east. That means the actors are actually walking in a northerly direction. The error is compounded by the actors/soldiers first muttering and then yelling "we are going south".
During the scenes depicting the Battle of Shiloh, the Union troops are shown with fixed bayonets on their muskets, but they're also carrying them on their belts.