How do we know the universe is expanding and wasn't already infinite? From what I know, the decision that the universe was expanding came from the discovery of the redshift and that other celestial objects, such as galaxies, were moving away from us; the farther they are, the faster they are moving. My question is, how was it decided that the universe is indeed expanding and not just infinite, as there is supposedly an infinite amount of space for it to expand. I'm also wondering how objects like galaxies move as a result of the expanding universe, as there are many that are currently moving away, I don't understand how it is possible for them to move.
 A: The expansion of the universe isn’t like a border that’s being moved; the universe could be positively curved, which means that if you go in a straight line long enough, you end up at the same place you started (thus realizing it wasn’t a straight line), like a globe, but now in four dimensions.
So this “globe” is expanding, and just as with a three-dimensional globe, there isn’t a border that’s being moved, the amount of area/volume (space in general) just increases between non-interacting bodies.
The universe could also be flat or negatively curved, it would be infinite from moment one, but it could still “expand”, which is rather just an upscaling of distance (as if you’re zooming in on a grid).
A: The universe, if it's either unique or multiversal, can be infinite and still be expanding.  Infinity's a concept, not a number, but any part of it can be represented numerically until you run out of ink molecules, or electrons, or whatever you're using to represent the numbers:  Either the infinity symbol or adjectives like "quasi-" or "AFAIK" can (at least for now) be used to indicate uncertainty about it.
The universe is supposed to be 13 or 14 billion years old, but the most recent unit of time in all those years can still  be subjected to infinite divisibility, and so can the oldest; also, in an inflationary multiverse, the age represents the "local universe", as the others are generally expected to remain causally separated, except in the extremely rare "bubble collisions" allowed by some models.
One attempt to prove a hypothesis that time may be an "emergent" phenomenon not always extant was Ekaterina Moreva's 2013 experiment, described on the 13th of the unnumbered pages of a preprint by Simenov that's visible at     https://arxiv.org/ftp/arxiv/papers/1505/1505.05724.pdf, but her results were not conclusive.
However, another experiment, conducted by Marco Genovese (a physicist at Turin's Instituto Nazionale di Ricerca Metrologica) and detailed on p.83 of Zeeya Merali's 2017 pop.-sci. book titled "Big Bang in a Little Room" (available, in the U.S., thru inter-library loans), provided more definitive results, using a two-photon model of a (very) local universe whose entangled polarizations and rotations left a direction of their passage through time evident to observers within it, while appearing to remain static to observers outside it.
As the question's 1st tag was cosmology, I should add that a cosmological model using that potential for divisibility which I've described is Nikodem J. Poplawski's "cosmoology with torsion", found by his name in Physics Letters A, in Cornell University's Arxiv website, and elsewhere on PSE.
A: Although this can be mathematically proven and was done so formally by Stephen Hawking and Roger Penrose it is actually intuitive to understand how the universe had a beginning in time. 
Notice that through numerous observations include redshifts of galaxies we have confirmed that the universe is expanding. This means that as time moves forward the space between every point in the universe increases(also note that this spacing is finite and the expansion of the space is also now at a finite rate). Thus it is logical to conclude that if we go back or rewind time then the universe is contracting(the opposite of expansion). Thus if we know that there must’ve been a time where that spacing between points (that I talked about), were zero as both the spacing and the rate of increase was spacing is finite. Thus this condition of space where there is no space between points is the supposed singularity we often hear and sure you may call it an infinite universe but it’s not accurate in the sense we imagine(with real space between points as no time can act if there is no change possible in space).
A: The short answer is Hubble's Law.
Astronomers have calculated the distance of objects from earth and found that they are accelerating away from us at speeds that increase proportionate to the further you look. So the "idea" is that these objects are not moving away faster.. but that space between them and us is actually growing larger.  Its a rough concept to visualize but you can make an analogy with swimming in a current in the ocean.  Imagine swimming at 1 mph.  If your swimming with a current of 1 mph, for one hour, then you would travel 2 miles in that one hour.  You however only swam for 1 mile.  Hubble's law says space is expanding.  The expansion has a similar affect on Galaxies that the current would have on you.
I have problems with Hubble's law because Time Dilation does a better job at explaining the reason for what appears to be "expansion."  But that's a different topic.  The current rate of expansion is a source of debate.  Plank studies pins the expansion rate at 41 miles per second per Megaparsec (Mpc). Hubble's Law pins this at 46 miles per second per megaparsec.  Its a bit of a bummer for the Law because none of the numbers appear to be balancing out to an acceptable error rate.
Galaxies are like balls in a bowling ally.  They continue to move in the direction they are thrown until they come into contact with an opposing force (The Pins).
Hope this helps.
