As to which battles- that seems an impossible question. There are small engagements that had huge sociopolitical consequences and huge bloody engagements that really didn’t change a thing in the big picture. And, of course, everything in between- imo it would be folly to try to list the ten or fifteen “most important” as none of them exist independent of the others.
Gwilym is absolutely right but I thought it would be fun to take a stab at it anyways.
The idea is to choose battles that played an important role in the overall progress of the war, meaning some smaller battles are included and classics (like Fredericksburg, Chancellorsville, and Shiloh) are not.
First Battle of Bull Run 7/21/1861
First major battle of the war
Battle of Mill Springs 1/19/1862
Secured Kentucky for the Union
Battle of Fort Donelson 2/11 - 2/16/1862
Along with Henry opened the Cumberland and Tennessee Rivers
Battle of Pea Ridge 3/7 - 3/8/1862
Drove Confederates from Missouri
Battle of Island Number Ten 2/28 - 4/8/1862
Opened Mississippi
Battle of Forts Jackson and St Phillip 4/18 - 4/28/1862
Led to capture of New Orleans
Siege of Corinth 4/29 - 5/30/1862
Major objective of 1862 western campaigns
Seven Days Battles 6/25 - 7/1/1862
Lee enters the field
Battle of Antietam 9/17/1862
Invasion of North repulsed; emancipation proclamation issued
Battle of Perryville 10/8/1862
Invasion of North repulsed
Siege of Vicksburg 5/18 - 7/4/1863
Along with Port Hudson opened the Mississippi
Battle of Gettysburg 7/1 - 7/3/1863
Second invasion of North repulsed
Second Battle of Battery Wagner 7/18/1863
Black troops
Battle of the Wilderness 5/5 - 5/7/1864
Grant takes command in the East
Second Battle of Petersburg 6/15 - 6/18/1864
Union fails to take Petersburg starting siege
Battle of Atlanta 7/22/1864
Major Confederate city falls
Battle of Mobile Bay 8/5/1864
Blockade strengthened
Battle of Fort McAllister 12/13/1864
End of March to the sea
Battle of Bentonville 3/19 - 3/21/1865
Last confederate gasp in west
Breakthrough at Petersburg 4/2/1865
Now we’re in the Endgame
Appomattox Courthouse 4/9/1865
Lee surrenders
This being said, in some ways it might be more helpful to just study campaigns in a broad sense. They attacked here to seize this objective and won. You can learn the ins and outs of the battles later but if you want to get a sense of the broad strokes of the war and how it was won and lost skip the details of individual battles and look at the broader movement of armies.
Justification for not including a few battles -
Fredericksburg, Second Bull Run, and Chancellorsville can be summarized as "a series of Union generals tried to beat Lee without success."
Shiloh was an attempt to stop the campaign towards Corinth that did not succeed.
While the Atlanta Campaign as a whole is interesting, since I was limited in the total number of battles I skipped to the end. The rest can be summarized as "Sherman slowly pushed Johnston back towards Atlanta."
Ditto Overland Campaign - I chose the Wilderness because it showed Grant's mindset.
I did not include Franklin and Nashville because, while they were important, they can be summarized as "Hood chose not to pursue Sherman into Georgia and had his army destroyed."
You get the idea....