For the Big Bang, there’s the observable fact that the universe itself is expanding, as differences between galaxies and other galaxies has been increasing via some unknown force, often considered to be the expansion of the universe aided by dark energy pushing things apart. There’s also the stretching of light between different galaxies when it reaches us. The light, though it’s from different times, doesn’t match to it’s original distance, and stretches in weird ways to sort of give us glimpses of different times in space at the same time, showing that these galaxies are moving in different directions and distances away from a sort of center, though that center has never been found, it’s effect from the center is clear. That center is supposedly where the Big Bang started from an infinitely hot and matter-filled point that exploded outwards (not literally, that’s for effect in my phrasing). There’s also microwave background radiation. Basically speaking, it’s the glow leftover from the extreme heat of the Big Bang, manifested in microwave background radiation, which we have mostly mapped and studied it’s existence, and have proven that Cosmic Background Radiation is true, real, and there, remaining one of the biggest pieces of evidence in the Big Bang theory. There’s also composition of stars by elements, which the Big Bang theory predicts that elements like helium and hydrogen where created in certain amounts towards the beginning of the universe. The amount and composition of these elements doesn’t apply to newer stars like the sun, but it does to far older ones, which we have empirically matched to the composition in the Big Bang theory. Alternate theories to the Big Bang theory, such as the Steady State Theory, hold that the universe has changed very little in the course of its history, but using telescopes such as Hubble or James Webb, we’ve proved that, as the Big Bang theory holds, the universe has experienced near constant change, and in large quantities overtime.