How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary cardinality? I'm sure I saw this result somewhere but I can't seem to find it anywhere (books, google,...) Thanks!
For any algebraic theory that is expressible in first order logic in a countable language, and this includes groups, rings, fields, partial orders, lattices, etc. etc., then the basic fact is expressed by the Lowenheim-Skolem theorem, which asserts that if the theory has an infinite model, then it has models of every infinite cardinality. In general, one gets models of the theory of every cardinality above the size of the language (and this covers your $R$-module case).
One needs the Axiom of Choice to prove this, however, and this use is necessary, since the Axiom of Choice is equivalent to the assertion that every set carries a group structure, as explained in the answer to this MO question.
I just wanted to point out that I don't think that the Axiom of Choice is necessary to
construct arbitrarily large groups. Can't you for each set $X$ take the collection of
all formal (finite) linear combinations of elements of $X$ over the rationals,
which then becomes a $\mathbb Q$-vector space of dimension the size of $X$?
Note that without AC we probably don't know that (in the case that $X$ is infinite) there is a bijection between $X$ and this vector space.