This paper investigates the first- and second-order maximum achievable rates of codes with/without cost constraints for mixed {channels} whose channel law is characterized by a general mixture of (at most) uncountably many stationary and memoryless discrete channels. These channels are referred to as {mixed memoryless channels with general mixture} and include the class of mixed memoryless channels of finitely or countably memoryless channels as a special case. For mixed memoryless channels with general mixture, the first-order coding theorem which gives a formula for the $\varepsilon$-capacity is established, and then a direct part of the second-order coding theorem is provided. A subclass of mixed memoryless channels whose component channels can be ordered according to their capacity is introduced, and the first- and second-order coding theorems are established. It is shown that the established formulas reduce to several known formulas for restricted scenarios.