Is every direct product of (finite) cyclic groups abelian?

Yes.

If $G\cong \langle a\mid a^m\rangle, H\cong\langle b\mid b^n\rangle$, then

$$G\times H\cong\langle a,b\mid a^m, b^n, ab=ba\rangle.$$

See here for details.

If you look at the way the direct product is defined, in particular the group operation, it's pretty easy to see that the answer is yes. For we have $(a,b)*(c,d)=(ac,bd)=(ca,db)=(c,d)*(a,b)$.

Thus every direct product of abelian groups is abelian.