A set is infinite iff (if and only if) it's the same size as at least one of its own proper subsets. (This amounts to saying that a set is infinite iff it's not finite.) A set is denumerable iff it's the same size as the set {1,2,3,...} of positive integers...or, equivalently, the same size as the set {0,1,2,3,...} of natural numbers.
Now the fact is that all the sets you identify are denumerable, and hence are all of exactly the same size. When you refer to "the set of numbers from 0 to infinity", I take it that you refer to what I just called the set of natural numbers. When you refer to the "set of even numbers from 0 to infinity", I take it that you refer to the even numbers among the set of natural numbers: 0, 2, 4, 6, ... . When you refer to "the set of prime numbers from 0 to infinity", I take it that you refer to the numbers 2, 3, 5, 7, 11, ... . These sets are all infinite, but they're all the same size: they're all denumerable, the same size as the set of natural numbers and the same size as the set of positive integers. These are all the same size because there's a 1-to-1 correspondence between the members of any one of these sets and the members of any other. Check it out.
Now it's true that there are infinite sets of different sizes, some larger than others. But you haven't gotten to them yet. All of the following are infinite sets of the same size as each other, but larger than any denumerable sets: the set of points on a line one inch (or a billion miles) long; the set of points in an infinite space of denumerably many dimensions; the set of distinct denumerably long strings of 0's and 1's; the set whose members are all the sets of positive integers; the set of real numbers from 0 through 1; the set of all real numbers, both positive and negative, including 0; and so on and on and on.
As I said, these sets, larger than the denumerable sets, are all the same size. There are still larger infinite sets than these, and infinite sets even larger than them, and so on ad infinitum.