Arguably the Napoleonic Wars. It was first instance in the history of warfare that you had battles fought by armies of 500,000 men and up. It was the first war to be fought on every single continent, expect Antarctica, and in all major oceans and it definitely affected all subsequent wars, not to mention global politics for a hundred years.
Of course if you're talking about numbers then a few battles in the Warring States Period of China (c.500BC - c.200BC) qualify up there - one state once mobilized every person above 15 in a single province to fight in a battle. And if you're talking about the breadth of conflict the Seven Years' War achieved similar scale. Not to mention that for most of the period the Napoleonic Wars were at a much lower level than that.
As for what is the first modern war... I would say that, in a logistics sense at least, it was probably World War I. Increasing complexity/number of weapons led to an exponential increase in ammunition expenditure, which meant that for the first time in warfare food was not the primary concern. In addition, the unprecedented number of men being concentrated on small fronts made it no longer possible to just 'live off the land' like previous wars (even the 1870-71 Siege of Paris saw Prussian troops creating ad-hoc farms behind the lines).
Technologically I would say that the Crimean War is just as good a contender as the ACW, since most of the
effects of the technology -like better communications leading to 'remote-control' operations from the rear, and the impact of public opinion on strategy - had already occurred in the 1850s. McClellan's observations with regards to the killing power of the Allied weaponry probably influenced his thinking when the American Civil War came along.
However, I do agree that the American Civil War combines most of the hallmarks of what we would consider a 'modern war'.