I was going about this by trying to come up with a bijection between this set and the set of all infinite sequences of zeros and ones (like 010100....), which I know in uncountable. But I didn't have much success. Other thoughts on how to get started?
It is enough to come up with an injection from the set of infinite sequences of zeros and ones into the set of complete and transitive relations.
So suppose you have a sequence like $(a_n)=(0,1,0,1,0,0,\ldots)$. You want to define a relation $R$ that depends somehow on $(a_n)$. You're going to say that $iRj$ if and only if... [something to do with $a_i$ and $a_j$]. Can you come up with a definition that makes $R$ complete and transitive? And is the mapping from sequences to relations injective?