Basically, there are a few known facts: (1) organisms die, (2) organisms reproduce, (3) the offspring are different from each other and from the parents, (4) the differences are more or less random, (5) dying is not random but a consequence of external forces, and (6) external forces are relatively constant. From these facts, it logically follows that over generations (or ofganisms reproducing ans dying), the living organisms will become more and more adapted to the environment. To avoid this long and detailed explanation, we just call this process evolution.
It's completely the same as e.g. the word "evaporate" - instead of saying that kinetic energy causes H2O molecules to become more and more excited, and slowly breaks the H bonds between molecules, so that they can move freely inspace (gas), we just say "water evaporates".
Does point (5) needs any experiments? While point (6) just needs a more extensive definition.
These 6 points are giving you the proof that you are asking for evolution to be proven in a scientific method.
Edit: fixed typos.
You don't need a definition of "species" to understand or work with evolution.
http://en.wikipedia.org/wiki/E._coli_long-term_evolution_exp...
"Evolutionism was a common 19th century belief that organisms inherently improve themselves through progressive inherited change over time, and increase in complexity through evolution. The belief went on to include cultural evolution and social evolution."
And more to the point:
"In modern times, the term evolution is widely used, but the terms evolutionism and evolutionist are seldom used in the scientific community to refer to the biological discipline as the term is considered both redundant and anachronistic, though it has been used by creationists in discussing the creation-evolution controversy. Apart from the use of the word by creationists, it is also often used by theists to imply a worldview that is essentially naturalistic."
In any case, your comment about evolving iguanas into cats and dogs indicate that you have a strange idea about about how evolution works. If a bad teacher gave you some wrong ideas then I can understand you skepticism. But you should probably try to understand evolution first before trying to criticize the theory as unscientific.