Yes, earth has both translatory as well as rotational motion. When the earth revolves around the sun it changes its position, that motion is translational in nature. But when the earth spins around its own axis, the motion is rotational motion. The earth revolves around the sun while spinning about its own axis, that's why the earth is known to have translatory and rotational motion both.
The motion of a wheel of a cycle is also an example when the body is in translatory and rotational motion both. The wheel of the cycle rotates about its own axis while it moves forward.