The first thing to note is that coffee was the hot beverage of choice. After the Boston Tea Party and general revolutionary sentiment got started, tea was mostly replaced by the newly christened Americans by coffee or "liberty tea", various herbal teas made from loosestrife, red sumac berries, raspberry leaves, goldenrod, and mint. Liberty teas fell out of fashion after the revolution, leaving coffee, chocolate, and tea (in smaller amounts than before) the main hot beverage for all classes. As Americans expanded westward, coffee went with them.

What all this means is that when the Civil War comes around, coffee is included in the rations. It's generally issued as green (fresh) beans that the soldiers would need to roast and grind themselves; without specialized equipment, this usually took the form of roasting the beans in a frying pan over the fire and grinding them in a cup with something like a bayonet socket. Not exactly the most precise brewing process, and without filters you just have to dump hot water over the grounds and try to use a stick or something to strain them as you pour. The Confederacy infamously ran short of coffee over its entire territory for both civilians and soldiers, and the Union would have shortages at times as well, leading to a lot of experimentation with roasted acorns, peas, barley, corn, chicory root, dandelion root, and other substitutes.

So if you have tea, it's likely a private purchase from a nearby town or a sutler following the army. Southerns enjoyed iced tea, but the Union blockades meant that tea was in short supply just like coffee. The dominant varieties of tea were much the same as that from the Boston Tea Party, both black and green varieties. Common examples include Bohea, Congou, souchong (not quite as smoky as common lapsang souchong, but still lightly smoked), and Hyson and Singlo green tea. All of these varieties can be purchased from retailers like this one. In 1859 you also started getting Japanese tea imports, but it wasn't the highest quality. I'd stick to the older Chinese stuff. For a Southerner, there wouldn't be anything unusual about reviving some of the old herbal teas to replace your coffee!

One tea that wasn't actually tea was "Beef Tea." This is a simple beef broth with all the fat skimmed off and a little pepper added if necessary, which was prescribed to hospital patients with fever.