What were the teams of the civil war?

There weren't any. Many soldiers in the Civil War saw baseball played for the first time, and learned to play themselves. Union General Abner Doubleday is said to have "invented" baseball, but what he really did was write down the rules for the game he saw his men playing. The first professional baseball team was created in 1869, four years after the war was over, the Cincinnati Reds. That was also the year the first College Football game was played. By the 1880s college football was big, but professional football did not start until after WWI, when the NFL was created in 1924. Basketball was invented in the 1890s, but the NBA did not start until after WWII, in 1948. At the time of the Civil War there was some boxing (mostly bare knuckle), some men liked to wrestle, and horse racing was popular.