Approaching my recommendations more analytically-- I think a good understanding of the riverine war would encompass these areas:
1. Overview. I would choose two here; for a 19th-century perspective, A.T. Mahan's Gulf and Inland Waters. And then for a more historical perspective, Milligan's Gunboats Down the Mississippi.
2. The opening battles in the northern reaches of the river. Cheairs, Battle of Belmont would be good here.
3. Forts Henry and Donelson. Cooling, Benjamin Franklin, III. Forts Henry and Donelson: The Key to the Confederate Heartland.
4. Island No. 10. Daniel and Bock's Island No. 10.
5. You might want to insert your favored treatment of Shiloh here. Although the impact of the gunboats at Shiloh was not huge, it's still a part of the larger story.
6. The capture of New Orleans. Hearn, Capture of New Orleans.
7. Fort Pillow and Memphis. McCaul, To Retain Command of the Mississippi.
8. Initial thrusts at Vicksburg. Bearss, Rebel Victory at Vicksburg, (and the first volume of Bearss' trilogy if you can find it).
9. Vicksburg. Ballard's Vicksburg is good, or whatever your favored treatment is.
10. Smaller actions. Jack Smith's Tinclads and his biography of Le Roy Fitch, Jr.
11. Red River. Joiner's One Dam n Blunder from Beginning to End would do well here.
12. In-depth on a particular gunboat. Smith's Carondelet.
13. Confederate angle. Campbell, Confederate Naval Forces on Western Waters (and perhaps Chatelain's Defending the Arteries of Rebellion, though I haven't had time to crack into it yet).
14. A civilian/Confederate angle. Gunther, Charles F. and Allardice, Bruce S. (ed.) Two Years Before the Paddlewheel: Charles F. Gunther, Mississippi River Confederate.
15. Perspectives from the sailors. Milligan's From the Freshwater Navy and Tomblin's Civil War on the Mississippi.
That should do it. I would imagine that by the time you get through all those, you'll know where you want to go next...
Oh... I should have thrown in Wideman's Sinking of the USS Cairo and Bearss' Hardluck Ironclad. Perhaps in between 8 and 9? Also, Browning's Lincoln's Trident treats Farragut's lower-river force in some detail and is worth a look. Donald S. Frazier's ongoing "Louisiana Quadrille" is excellent for the regions of the lower river but is too in-depth for the first go-round... look at it after you have the basics down.