Suppose there is a line, infinitely long in both directions. Make arbitrarily "uniform" cuts or "integers". Obviously there are infinitely many of these. And there are arbitrary "lengths" BETWEEN these units. Now, there are infinitely many WAYS("lengths") to chop each of these lengths into units. So we can see that there are different TYPES of infinities here - the AMOUNT of cuts vs WAYS of cutting BETWEEN those cuts.
So perhaps, it's not that there are different SIZES of infinity, but it's just that you are counting different things when you compare the "sizes" of, say, integers vs real numbers?
No. The "number" of integers has nothing to do with how they are laid out on a number line. On a line, the point labelled $0$ and the point labelled $1$ forces all of the other integers to correspond to well-defined, unique positions on that line. There is a very clear one-to-one correspondence between the "integers" of any two such lines and ways and types are just bothersome noise.