What evidence do you have that plants and animals live long ago?

There are now mountains of evidence in support of the fact of evolution (it's as much a fact as the existence of atoms, or the idea that disease is caused by pathogens), as well as for the Theory of Evolution by Natural Selection. The evidence comes from five independent areas:

• Biogeography: distribution of species on the planet. For example, there are loads of birds and plants on volcanic islands around the world, but frequently no, or very few, mammals. The explanation is that after a volcanic island has formed in the middle of the ocean, it's far harder for mammals to reach it than for birds and plants, which can be driven on the wind, or can float very easily.
• Embryology: for example, dolphin embryos clearly show signs of hind legs, but these disappear before birth.
• Vestigial and atavistic organs: horses and rabbits manufacture their own vitamin C. Humans do not. However, humans do have most of the equipment for doing so, but it is non-functional because the genes for the equipment are somewhat broken.
• Genetic/molecular evidence (DNA): It is clear that there is a hierarchy of descent and relatedness when we compare DNA across many, many species.
• The fossil record

Note that we could entirely discard the fossil record, really, throw it all away, and the other four areas of support would still be overwhelmingly positive. Even without the fossil record, we can accept evolution as a fact. So if you ever hear someone challenge evolution or evolutionary theory based on some supposed flaw in the fossil record, just let them discard the entire fossil record, just for the sake of argument. The evidence is still overwhelming.

See the wonderful lecture by evolutionary geneticist Jerry Coyne in the related links for this question. If you are religious, or easily offended, you might want to steel yourself, as his frustration with creation science and Intelligent Design is very evident. Still, if you can look past his lack of diplomacy, there is a wealth of good information in the lecture. He discusses not only the copious evidence for evolution, but also the nature of scientific fact and theory. The lecture is based on his book, Why Evolution is True.