If you say one kind of infinite is smaller than the other kind, then the first kind no longer qualifies to be called as an infinite as it smaller than some other number. So first you need to define what an infinite is.

Also infinite is not a number. And comparison exists only for numbers.

The finite numbers extend easily to "cardinal numbers", which may be infinite: https://en.m.wikipedia.org/wiki/Cardinal_number

This is not right.

There are infinitely many integers.

There are infinitely many real numbers between each pair of integers.

Thus there are more real numbers than integers.

Unfortunately this line of argument doesn't quite work either. You could replace "real numbers" by "rational numbers" and it would still be true except for the last line. The size of the integers is the same as the size of the rationals. You have to think in terms of injective functions.