Help with denoising signal and periodogram analysis resources
Firstly, I am confused if I am supposed to filter my signals to get rid of any frequencies above the Nyquist frequency. My sampling frequency is 32Hz and my time series is somewhat noisy and has some artifacts. I am also unsure of which filter to select for this.

That ship has sailed.
Let $S=\left\{\left.\alpha e ^{i(\omega t+\varphi)}\right|\alpha &gt; 0, 0\le \varphi &lt;2\pi, \omega \in \mathbb R\right\}$, i.e. the set of all distinct complex sinusoid with an amplitude, frequency and phase.
Then $(S,\cdot)$ is a commutative semigroup (proof trivial).
Introducing the equivalence relation $\sim: a\sim b \iff a\left(\frac{n}{r}\right)= b\left(\frac{n}{r}\right) \forall n\in\mathbb Z$ (&quot;two signals are identical after sampling with rate $r$&quot;), we see that signals $s_l=\alpha e^{i(\omega_l t + \varphi)}, l=1,2,\ldots$ are $s_1\sim s_2$ if $\frac{\omega_1-\omega_2}{2\pi}=nr, n\in\mathbb Z$, i.e. we can't tell signals apart after sampling if their frequency differed by a multiple of the sampling rate.
Let's formalize this: $T=\left\{\alpha e^{i([ft+\phi]\mod 2\pi)}\right\}$ is a quotient semigroup of $S$, i.e. $T\preceq S$, and each of the elements is a leader of a $\sim$-equivalence class – and this homomorpism $S\mapsto T$ is in fact sampling, which, as we can see above, is not bijective.
Hence, all the original frequency components from $S$ got mapped to some component from $T$ with frequency normalized to the sampling frequency. That mapping is called aliasing.
What an anti-alias filter $h$ does is
$$h(s): S\mapsto S, h=\begin{cases}
s, &amp; f&lt; f_\text{nyquist}\\
0 &amp; \text{else,}
\end{cases}
$$
and as you'll figure out when inserting $h(s)$ above is that this yields only the elements that are not aliased to a different frequency.
Thus, everything that &quot;survives&quot; $h$ will also &quot;survive&quot; aliasing without undergoing a change in frequency.
So, if you needed an anti-aliasing filter, it's now too late. Go and do your recording again.

Secondly, I wanted to look at the periodogram and the average power spectral density at each frequency within a group - but I am not sure if I understand the periodogram very well - if I have different time series lengths then my periodogram length will vary too, so I am not sure how this comparison can be made.

In that case, the periodogramm is not a useful mapping on its own – you'll need to add something like a truncation / padding operation to bring all signals to the same duration, for example. At which point the periodogram doesn't seem to be a sensible approach anymore.