Currently there is almost no doubt in the scientific community that the Universe is, in fact, expanding.
Cosmic Microwave Background Radiation
Some more evidence for the Big Bang was discovered in the 1960's. By pointing a radio telescope to the sky, we could practically "hear" the "echo" from the Big Bang. Instead of being a sound, it is actually leftover thermal radiation from the Big Bang which permeates the entire Universe.
At the moment just after the big bang, the universe was full of tremendously hot hydrogen plasma emitting radiation at all frequencies. As the universe expanded the gas thinned out. Eventually it went from glowing white hot, to red hot, then continued to cool as it expanded so that today it is so cold that it only emits gentle microwaves.
The thing is that these microwaves are at the exact intensity that the big bang theory predicts they should be given the age of the universe. No other theory successfully predicts the nature of the microwave background, so that is one of the many reasons scientists, as well as all scientifically literate folk, choose to accept the big bang theory as the best theory that we currently have to explain the current state of the universe.
The fact that this radiation is detected within the same ranges in all directions confirms that it doesn't come from a fixed or finite source, but rather from the early universe itself.