How many relations are there between the set A and B?

Yes, you did. There are $3 \cdot 2=6$ pairs of one element from $A$ and one from $B$. Each of these pairs can be in the relation or not, so you have six twofold choices that are independent. That gives $2^6=64$

The total number of relations that can be formed between two sets is the number of subsets of their Cartesian product.

For example: $$ n(A) = p\\ n(B) = q\\ \implies n(AXB) = pq\\ Number\ of\ relations\ between\ A\ and\ B = 2^{pq}\\ $$

Remember that if $n(T) = m$, then the number of subsets of set $T$ will be $2^{m}$