How old is the earth?

4.566 billion years, plus or minus a few million years.

A nice accessible article on the issue is by Bob White, Professor of Earth Sciences at Cambridge and Director of the Faraday Institute. He reviews the science behind this value, and discusses why it should be of no concern to Christians.

The key issue in the science is that there are many independent measuring techniques that give the same answer.

How old is the universe?
13.8 billion years plus or minus about 40 million years.

Again the key to my confidence in the answer is that there are several independent means of measuring the age of the universe:
* rate of Hubble expansion of the universe
* cosmic microwave background radiation
* relative abundance of light elements and isotopes [hydrogen, deuterium, lithium,..]
All give the same answer [within a one per cent].
It is only the last 20 years that enough precise data has been obtained to pin down all these estimates.