Let $G$ be the Prufer $p$-group. Then my question is, does $G$ contain subgroups which are isomorphic to a finite or countable direct sum of finite cyclic groups?
Of course, some direct sums of cyclic groups are themselves cyclic groups, but I'm interested direct sums of cyclic groups which are not cyclic.
No. As stated on Wikipedia, every proper subgroup of the Prufer $p$-group is finite cyclic. The Prufer $p$-group itself also is not a nontrivial direct sum (of any abelian groups whatsoever), because it has no nontrivial idempotent endomorphisms (its endomorphism ring is the $p$-adic integers).