For $p\neq 2$ the answer is yes. It's an easy computation usinghomology decompositions in the sense of Eckmann–Hilton the well known exact sequence
$$\operatorname{Ext}(A,\pi_{n+1}(X))\hookrightarrow [M(A,n),X]\twoheadrightarrow \operatorname{Hom}(A,\pi_n(X)),$$
and the computation
$$\pi_{n+1}(M(A,n))=A\otimes\mathbb{Z}/2.$$
Here $M(A,n)$ is the Moore spectrum whose homology is the abelian group $A$ concentrated in degree $n$.
As you point out, it is easy to check that
$$H_n(M(A,s)\wedge M(B,t))=
\begin{cases}
A\otimes B,&n=s+t,\\
\operatorname{Tor}_1(A,B),&n=s+t+1,\\
0,&\text{otherwise}.
\end{cases}$$
Therefore, $M(A,s)\wedge M(B,t)$ can be obtained as the homotopy cofiber of a map
$$f\colon M(\operatorname{Tor}_1(A,B),s+t)\longrightarrow M(A\otimes B,s+t)$$
which is trivial in homology $H _{*}(f)=0$.
Suppose for instance that $A$ and $B$ are finite and do not have $2$-torsion. Then the previous short exact sequence shows that homology induces an isomorphism
$$H _{s+t}\colon [M(\operatorname{Tor}_1(A,B),s+t), M(A\otimes B,s+t)]\cong \operatorname{Hom}(\operatorname{Tor}_1(A,B),A\otimes B).$$
Therefore $f$ must be null-homotopic, so
$$M(A,s)\wedge M(B,t) \simeq
M(A\otimes B,s+t)\vee M(\operatorname{Tor}_1(A,B),s+t+1).$$
If you take $A=\mathbb{Z}/p^i$ and either $B=\mathbb{Z}/p^j$ or $B=\mathbb{Z}/p^i$ you always obtain the same thing on the right.
For $p= 2$ the answer is no. The spectrum $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ is the mapping cone of
$$4\colon M(\mathbb{Z}/2,0)\longrightarrow M(\mathbb{Z}/2,0)$$
which is knonw to be null-homotopic, actually $[M(\mathbb{Z}/2,0),M(\mathbb{Z}/2,0)]\cong\mathbb{Z}/4$ generated by the identity. Hence
$$M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)\simeq M(\mathbb{Z}/2,0)\vee M(\mathbb{Z}/2,1).$$
In particular the action of the Steenrod algebra on the mod 2 homology of $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ is trivial.
On the other hand, it is known that the Steenrod algebra mode 2 acts on the mod 2 homology of $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/2,0)$ in a non-trivial way, i.e. the first Steenrod operation sends the 0-dimensional generator to the 1-dimensional generator. Therefore $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/4,0)$ and $M(\mathbb{Z}/2,0)\wedge M(\mathbb{Z}/2,0)$ cannot be homotopy equivalent.
As for references, see Hatcher's book. This book doesn't deal with spectra but since we are working with finite-dimensional CW-complexes you can assume that we are in the stable range.
With respect to the first question, expanding on my comment which pointed out the nLab page dependent linear type theory and the article by Urs Schreiber, 'Quantization via Linear homotopy types', I'd like to pass on some additional relevant commentary by Urs.
There is a way to express linearity for homotopy types, see stable homotopy type. As Simon pointed out in his comment, the category of parametrized spectra is an ∞-topos, indeed a tangent ∞-topos. One can then look to add an axiom to HoTT that makes it the internal language for tangent ∞-toposes. That axiom should be:
there is a pointed type $X_\ast$
the morphism $\Sigma \Omega X_\star \to X_\star$ is an equivalence
So far this makes the ambient category "linear" in the terminology of Schwede's old article on parameterized spectra. One could now continue to add the axioms
- the morphism $\Sigma \Omega(X_\star^{S_\star}) \to X_\star^{S_\star}$ is an equivalence, for finite pointed powers.
If one could add the infinite set of these axioms, that would be Goodwillie's localization for the tangent $\infty$-topos.
There's work in preparation by Mathieu Anel, Eric Finster, and André Joyal along these lines.
Best Answer
There have been several questions previously in this vein. This one asks for an advanced beginners book. The consensus seemed to be that it was difficult to find a one-size-fits-all text because people come in with such diverse backgrounds. Peter May's textbook A Concise Course in Algebraic Topology is probably the closest thing we've got. If you like that, then you can also read More concise algebraic topology by May and Ponto. I also recommend Davis and Kirk's Lecture Notes in Algebraic Topology. I think these would be a very reasonable place for a beginning grad student to start (assuming they'd already studied Allen Hatcher's book or something equivalent).
Another question asked for textbooks bridging the gap and got similar answers. Finally, there was a more specific question about a modern source for spectra and this has a host of useful answers. Again, Peter May and coauthors have written quite a bit on the subject, notably EKMM for S-modules, Mandell-May for Orthogonal Spectra, and MMSS for diagram spectra in general. Another great reference is Hovey-Shipley-Smith Symmetric Spectra. On the more modern side, there's Stefan Schwede's Symmetric Spectra Book Project. All these references contain phrasing in terms of model categories, which seem indispensible to modern homotopy theory. Good references are Hovey's book and Hirschhorn's book.
Since you mention that you're especially interested in $E_\infty$ ring spectra, let me also point out Peter May's survey article What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?