Let $p$ be a prime and $(p, n)=1$. I am wondering how to explicitly compute the norm groups of $\mathbb{Q}_p[\zeta_n]/\mathbb{Q}_p$. Ideally, I would like a computation using class field theory and also an elementary one.
Using local class field theory, we get an isomorphism $\mathbb{Q}_p^\times/N(\mathbb{Q}_p[\zeta_n]) \cong \operatorname{Gal}(\mathbb{Q}_p[\zeta_n]/\mathbb{Q}_p)\cong \operatorname{Gal}(\mathbb{F}_p[\zeta_n]/\mathbb{F}_p)$, and this group should be cyclic with order equal to the order of $p\pmod n$; call this number $k$. Then we are looking for a subgroup of $\mathbb{Q}_p^\times \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p \times \mathbb{Z}$ with quotient group $\mathbb{Z}/k\mathbb{Z}$; it is very likely given by $\mathbb{Z}_p^\times \times \langle p^k\rangle$ although some argument may be needed to show it is not a different subgroup that gives an isomorphic quotient.
But also, I would like a computation of this norm group without appealing to class field theory.
When $(p,n) = 1$, $\mathbf Q_p(\zeta_n)$ is an unramified extension of $\mathbf Q_p$.
When $K/\mathbf Q_p$ is unramified, a prime in $\mathbf Q_p$ is also prime in $K$. Using $p$ as a choice of prime, we can write $$ K^\times = p^\mathbf Z \times \mathcal O_K^\times, \ \ \ \mathbf Q_p^\times = p^\mathbf Z \times \mathbf Z_p^\times. $$ Let ${\rm N} \colon K \to \mathbf Q_p$ be the norm map. Check ${\rm N}(p) = p^d$, where $d = [K:\mathbf Q_p]$. (You wrote the degree as $k$, which is weird when your top field is denoted $K$, so I'm not going to use that.) Check ${\rm N}(\mathcal O_K^\times) \subset \mathbf Z_p^\times$.
Key point: when $K$ is unramified over $\mathbf Q_p$, ${\rm N}(\mathcal O_K^\times) = \mathbf Z_p^\times$. Proving that requires some work, and it is relies on surjectivity of the norm map $\mathbf F_{p^d}^\times \to \mathbf F_p^\times$ and trace map $\mathbf F_{p^d} \to \mathbf F_p$ on finite fields.
Class field theory expresses the index of the image of the norm map on units in terms of the ramification index, so that's how you use local class field theory to show ${\rm N}(\mathcal O_K^\times) = \mathbf Z_p^\times$ when $K/\mathbf Q_p$ is unramified.
The unramified case is the easiest one. It's worth being able to work this out from scratch in a direct way and by local class field theory, but computing norm groups for ramified extensions is much more complicated, so feel free to rely on local class field theory to help you make the calculation in ramified examples. Although I guess it's also worth trying to directly make the calculation for ramified quadratic extensions (when the residue field characteristic is odd).