Can a torsion group and a nontorsion group be elementarily equivalent? I think that the direct product
$\prod_{n \in {\bf N}\setminus \{0\}} {\bf Z}/(n)$
and the direct sum
$\bigoplus_{n \in {\bf N} \setminus \{0\}} {\bf Z}/(n)$
are elementarily equivalent but am not sure how to prove it. 

Yes, a torsion group can indeed be elementarily equivalent to a non-torsion group. 
We can get a (somewhat) explicit example using the following fact, together with Łoś's theorem:

If $G$ is torsion, but has elements of arbitrarily large finite order, then any nontrivial ultrapower over $\mathbb{N}$ of $G$ is non-torsion.

Proof: suppose $G$ is as above, and $\mathcal{U}$ is a nonprincipal ultrafilter on $\mathbb{N}$. Then let $a_i\in G$ have order $>i$, and consider the element $$\alpha=[(a_i)_{i\in\mathbb{N}}]_\mathcal{U}$$ of the ultrapower $\prod_{\mathbb{N}}G/\mathcal{U}$. For each $i$, the formula "$x$ has order $>i$" holds of cofinitely many $a_j$s (namely, each $a_j$ for $j\ge i$); since $\mathcal{U}$ is nonprincipal, every cofinite set is in $\mathcal{U}$, so $\alpha$ has infinite order.
As usual, this ultrapower argument can be replaced by a simple compactness argument - see Alex Kruckman's answer. I gave the ultrapower construction above since I think it's cool.

I believe that in particular, the groups you mention in your question are elementarily equivalent, but I don't immediately see how to show that (I suspect a proof via Ehrenfeucht-Fraïssé games wouldn't be too hard though).
