The explicit calculation of this Heegaard-Floer homology (at least, $HF^-$, and of the unique $\text{Spin}^c$ structure for which $c_1(\mathfrak s) = 0$) was carried out in the paper in which it was introduced, and indeed is an important part of the proof that Heegaard-Floer homology is well-defined. See Lemma 9.1. One also has the isomorphisms $HF^+ \cong \Bbb Z[U,U^{-1}]/U\Bbb Z[U] \otimes H_*(T^g;\Bbb Z)$ and $HF^\infty \cong \Bbb Z[U^{-1},U] \otimes H_*(T^g;\Bbb Z)$. The exact sequence relating the triple ($HF^+, HF^-, HF^\infty)$ is exactly what you think it is. When the $\text{Spin}^c$ structure is nonzero, all the Floer groups are zero. (These four groups are all computable by hand, as done in the original paper.)
In this case, the Seiberg-Witten Floer homology is computable, because $S^2 \times S^1$ has positive scalar curvature, as do all connect-sums with itself by Gromov-Lawson. When $c_1(\mathfrak s)$ is not torsion (in this case, not zero), all the Floer groups are trivial, and in the case that $c_1(\mathfrak s) = 0$ all the Floer groups are explicitly calculable, as is the exact sequence (one of the connecting maps is zero) - for the precise result, see Chapter 36 of Kronheimer-Mrowka, "Monopoles and 3-manifolds".
To compare these results, one should note the relationship (now proved, thanks to Colin-Ghiggini-Honda and Kuthulan-Lee-Taubes) $\overline{HM} \cong HF^\infty$, $\check{HM} \cong HF^-$, $\widehat{HM} \cong HF^+$. $\widehat{HF}$ corresponds to a reduced Seiberg-Witten Floer homology not described in that chapter of Kronheimer-Mrowka, $\widetilde{HM}$.
