How fast is space expanding? Does the rate of expansion have any direct effect on the Earth or our galaxy?

The increasing rate of expansion will ultimately cause the heat death of the universe. As the universe expands the level of useable energy and matter will decrease (entropy will increase) Eventually there will be no useable energy or matter in the universe. This includes the earth and our galaxy, however there is no immediate physical effect on the earth.

However there is a mental philosophical impact. The philosophy that matter is eternal and self existent should have died with the discovery of the increasing rate of expansion of the universe. ie The previous idea that the universe would recycle between periods of expansion and contraction, ( Big Bang, and Big Crush) was proven false.

Philosophies of Science are still adjusting to the concepts that the universe is not eternal, the universe is an open system, and there might be something that exists besides matter and energy.

Since local gravitational forces oppose the effects of space expansion on small scales, the rate of space expansion is measured by the Hubble constant, which is currently estimated to be approximately 73.3 km/s per megaparsec. On cosmic scales, however, the expansion has an impact on the large-scale structure and ultimate destiny of the universe.