The Dold-Thom construction of a connected CW-complex $X$ is a product of Eilenberg-Maclane spaces so its homology groups are those of a product of Eilenberg-Maclane space which are determined by a Kunneth formula.
One basic example to see is the Dold Thom construction of a circle which is homotopy equivalent to a circle and also the examples of spheres more generally.
Of course, the Dold-Thom construction which is also called the infinite symmetric product construction $SP^{\infty}$ is a homotopy functor, meaning that we can compute the integral homology of infinite symmetric products of "simpler" spaces if these spaces are homotopy equivalent.