Intelligent Design, also known as Biblical Creationism, is readily disproved. Whether or not there is a god or not has nothing to do with whether or not a fact about the world can be proven or not. Most people are unaware of the errors in Darwin's books, which is ironic. As creationists are so anti-evolution, they should be pouring over his books to find the mistakes. The way we know he made mistakes is that science doesn't treat anything as sacred and tests everything.
The reason I bring up Darwin's errors is that showing they are false also happens to disprove creationism as well.
In the 1920's there was a geneticist by the name of RA Fisher. Fisher had a terrible dilemma. He was sure Darwin was wrong and Gregor Mendel was correct, but had no clear way to show this. There were others he also lumped into the "wrong" category. He then took a cue from Aristotelian logic, which you were likely taught in high school as truth tables, and devised a clever test. This test, when repeated, proves that creationism is false. It also disproves intelligent design as well, but intelligent design didn't exist in the 1920's.
There is a concept called modus tollens in logic. It is operationalized in high school math as "If A then B" and "Not B" therefore "Not A."
The A is called a null hypothesis and it is in particular called the "no effect" hypothesis. You assume the opponent is the only one who is correct and that everything you believe is false. This, of course, creates a giant mountain you have to climb if you start off with you being false as the given truth.
Fisher assumed that Mendel's laws have no effect and that ANY other explanation including intelligent design and creationism and Darwin's ideas and Lamarck's ideas could be the true one. All other ideas were in the "possibly true" category and Mendel's laws were in the "perfectly false" category.
Well, the neat thing about assuming Mendel's laws are false is that you can test that. To do so, their falseness has to create a prediction about how the data must appear if they are false. If the data appears far away, it may still be true, but that chance events caused you to grab a weird set of data. So Fisher devised away to calculate a worst case statistical distribution to account for the worst possible case of chance.
He then tested the data and the null hypothesis was excluded by the data alone. The data excludes everything except evolution.
Every time a biologist tests evolution, they do so by assuming evolution is the false idea and the data rejects this idea that evolution is false.
It doesn't matter if there is a god, it is nearly perfectly certain, given 150 years of data, that evolution is absolutely true.
Using the same tools you can, and biologists have, disprove intelligent design as well.
The nice thing is that you don't have to "research" any of this. You can do these experiments for yourself. Mendel grew peas. Undergraduate biology students regularly do tests like this. If you watch tv shows like CSI, you are watching tests built on evolution.
The Big Bang is an entirely different thing. And a post on it on top of evolution would be too long, but the way you can tell the Big Bang is true is two-fold. First, if you tune a radio to a particular set of frequencies, you can hear the echo of that initial rapid expansion. You would have to look it up. It sounds like the static on the radio between stations. The second is that if you use GPS to travel a great distance, then you are trusting Relativity, of which the Big Bang is part of.
If you drive a car from Maine to Kansas and get to where you are going, then you can trust the Big Bang.