I mean can we say for example omega squared corresponds to aleph one or something? Are aleph numbers and infinite ordinals completely different? Do cardinals and ordinals become two different sets of numbers when we think about infinity?
They're completely different.
Assuming the axiom of choice, and using the standard definition of cardinality in that case, every cardinal is an ordinal. But most ordinals are not cardinals. For your specific example, $\omega^2$ is countable but $\aleph_1$ is uncountable, so they can't biject.
Ordinals count the lengths of well-orders; cardinals count the sizes of unordered sets. Since there are lots of different orders that can be placed on any given set, there are correspondingly many different ordinals of a given cardinality. (There are uncountably many countable ordinals!)
Note that if choice fails, then ordinals and cardinals become even more completely different (though the alephs are still the same ordinals as ever they were - there are just more cardinals than the alephs alone); now, cardinals need not be ordinals at all, since not every set needs to biject with a well-ordered set.
Cardinal numbers (א numbers) are more about the size of the set, rather that what is contained within the set (ordinal numbers). There could not be a bijection a second that would be the same as comparing rocks to minutes or dollars to meters, it just doesn’t make sense.