The United States participated in the "Northern Front" of the war against the Japanese in Burma. Although the majority of the soldiers in that theater were from British India, Britain or other parts of the British Empire, the Northern Front was almost exclusively Chinese and American.

It might sound strange to someone casually interested in history to know that American and Chinese troops fought side by side under the command of an American general when less than a decade later, the United States and China would find themselves at war in Korea. In the case of Burma, that's exactly what happened. When the Japan invaded Burma in 1942, one of the primary objectives was to cut any and all land routes between India and China so as to prevent supplies from reaching Chinese nationalist leader Chiang Kai-shek and the armies under his command fighting the Japanese. After the successful invasion of Burma, the land route to China ("The Burma Road") was effectively cut off and the only way to get military supplies to China was through "The Hump", the air route between India and China that you mentioned in your question. Throughout the war, thousands of Americans were stationed in India to work on this airlift but also to cooperate with the British Empire where American planes became vital for the ever so important air supply that was one of the core tenants in the military strategy of the British command.

But we're not going to talk about the pilots, mechanics and all other Americans who worked with the airlift to China. You asked about the United States Army, and I will tell you about the US Army in Burma.

With the Burma Road out of order, a new road had to be constructed. Therefore, the primary military objective for the United States in Burma was to capture northern Burma and in the process build a new land route to China. To accomplish the military objectives, American general and commander of the Northern Combat Area Command Joseph "Vinegar Joe" Stilwell had three Chinese divisions and one American brigade, the subject of this answer: the 5307th Composite Unit (Provisional) or as they are most famously known as, Merrill's Marauders.

The Marauders were far from typical or average soldiers. Today we'd recognize them as elite light infantry soldiers and at the time, they were comparable with the US Ranger battalions or Marine Raiders. This also gives you an idea of what these men were meant to do: highly offensive operations behind enemy lines. They were originally only meant to be in combat for 3 months. Instead, they were in the field for 8 months in 1944. The almost 3000 men (plus an additional 2000 replacements) fought almost non-stop in one of the most gruelling campaigns of the Second World War against both the difficult challenges of the Burmese terrain and their Japanese foes. At the end, the Marauders triumphed in the capture of Myitkyina together with other elements of the NCAC. By that time however, only one-third of the Marauders were still able to fight - and even those remaining were in a bad shape. on August 8 1944, the Marauders were disbanded. They were replaced by the 5332d Brigade (Provisional), better known as MARS Task Force, that consisted of more than 7000 men.

How did a more or less normal day look like for a soldier in the Merrill's Marauders? It's always difficult to generalize because what really is a normal day during such extraordinary times? What is certain is that a normal day would involve marching. The Marauders were constantly on the move. They would march from one objective to the other through some of the most terrible terrain you can imagine. The day would start early in the morning and continue past the night unless the day's march had reached its destination. Between those hours, a soldier had to fight against two enemies: the Japanese and the environment. Soldiers had to keep a constant vigilance so as to not find themselves ambushed by Japanese soldiers or to attract their attention. Occasionally the Marauders would themselves ambush Japanese parties, often walking on trails. Yet the major combat that the Marauders would face were at their final objectives. Far more serious for a Marauder in the field was his immediate surroundings. Due to the difficult jungle terrain that the soldiers had to move through, they could not be supplied beyond what they carried on them, on their mules or were air-dropped to them. If they ate all their available rations before the next air-drop, they had to spend days hungry on the march. Simply marching in this environment was difficult in itself, but it became even more difficult if you were sick. The most common and serious sickness was amoebic dysentery (which 503 out of the 2600 original Marauders were inflicted with) but other diseases, some less serious, still plagued men as they marched. Unlike soldiers elsewhere, you couldn't be sent back for hospitalization and soldiers had to wait to be evacuated by air, something which only happened if they reached a potential airfield. Until then, you had to march. It is perhaps to no one's surprise that the Merrill's Marauders were shattered by the time their final objective, Myitkyina, had fallen. Exhausted, sick and suffering from malnutrition after facing the Japanese in irregular and conventional combat and having marched nearly 1000 miles - the majority of the soldiers ended up hospitalized.

That was the end of the Merrill's Marauders, but other units kept the fight going at the same time or after they left Burma. MARS Task Force being one of them, but the American participation in the Burma campaign contained many other units worth mentioning, ranging from the 1st Tank Group that supported Chinese divisions to Engineer combat battalions who also faced the Japanese. I cautioned before about generalizing and although the Marauders might be the most extreme of examples, they still give a good image of the realities of fighting in the Burmese jungles that the Chinese and British Imperial troops also had to face.

Sources:

A great overview for a new reader in the Burma campaign is The Burma Campaign: Disaster into Triumph 1942-45 by Frank McLynn (2011). For the Merrill's Marauders, I would point towards Spearhead: A Complete History of Merrill's Marauder Rangers by James E.T. Hopkins & John M. Jones (2000) and Merrill's Marauders: The Untold Story of Unit Galahad and the Toughest Special Forces Mission of World War II by Gavin Mortimer (2013). A smaller and highly illustrated booklet is Merrill's Marauders by Edward Young (2012).