I have just started studying Witt vectors and I have questions about the following identity $$W_n(mathbb{F}_p)cong mathbb{Z}/p^nmathbb{Z}$$

I would like proving this by finding an explicit map $phi_n: W_n(mathbb{F}_p)to mathbb{Z}/p^nmathbb{Z}$, but couldn’t come up with a reasonable map. Although I have found the isomorphism $$phi: W(mathbb{F}_p)to mathbb{Z}_p$$ via $(a_0,a_1,…)mapsto chi(a_0)+chi(a_1)p… $, where $chi$ is the Teichmüller character.
Can I obtain $phi_n$ by composing $phi$ with $pr_n:mathbb{Z}_p to mathbb{Z}/p^nmathbb{Z}$ and identifying $W_n(mathbb{F}_p)$ with $(a_0,…,a_{n1},0,0,…)in W(mathbb{F}_p)$? Is there a nice explicit version of this map?

More generally I am interested in the case $A=mathbb{F}_q$. I know that $W(mathbb{F}_q)cong mathbb{Z}_p[mu_{q1}]$ should hold.
Is it possible to argue that this is an isomorphism because $mathbb{Z}_p[mu_{q1}]$ and $W(mathbb{F}_q)$ are both strict $p$ring with residue field $mathbb{F}_q$ and as such canonically isomorphic?
Ad 1) First of all I’m pretty sure that yes, $phi_n = pr_ncircphi$, although usually the index is off by one (so I would have expected $W_{n1}(Bbb F_p) simeq Bbb Z/p^n$).
To make that map a bit more explicit, I remember a MathOverflow post which might be helpful here. It motivates the Witt polynomials (noted $W(X_0, …, X_{n1})$ there), which, if I’m not mistaken, are “more or less” your map $phi_n$ (annoying index shift again, oh well …). “More or less” because you have to choose some lifting
$$Bbb F_p rightarrow Bbb Z/p^{n}: quad bar a mapsto a.$$ But well, if you just go all the way up to $Bbb Z$ and work with the old school representatives ${0, …, p1}$, — and also remember that in this very special case, the Frobenius $(cdot)^{,p}$ is just the identity on $Bbb F_p$ –, you can explicitly write the map you call $phi_n$ as
$$(bar a_0, bar a_1, …, bar a_{n1}) mapsto (a_0^{p^{n1}} +p cdot a_1^{p^{n2}} + … + ,p^{n1} cdot a_{n1}) +p^nBbb Z$$
To make it totally explicit, say $p=5, n=3$, and say your element in $W_3(Bbb F_5)$ is $(bar 1, bar 4, bar2)$, it gets mapped to $$(1^{5^2}+5cdot 4^5 + 5^2cdot 2) +5^3Bbb Z = 5171 + 5^3Bbb Z = 46 + 5^3Bbb Z$$
— and note that crucial and fun fact that the choice of the lifting does not change the result; if e.g. instead you lift $bar 1$ to $6$, $bar 4$ to $9$, and $bar 2$ to $22$, you still get
$$(6^{5^2}+5cdot 9^5 + 5^2cdot 22) +5^3Bbb Z = 28430288029929997171 + 5^3Bbb Z= 46 + 5^3Bbb Z$$
Added: Let’s see the connection to the first part of Jyrki Lahtonen’s comment: one can get those Teichmüllerlike representatives of $Bbb F_p$ in $Bbb Z/p^n$ by raising any set of representatives to the $p^{n1}$th power; continuing the example above we get $chi(1) =1$ and
$$chi(2) = 2^{5^2} + 5^3Bbb Z = 33554432 + 5^3Bbb Z = 57 + 5^3Bbb Z$$
$$chi(3) = 3^{5^2} + 5^3Bbb Z = 847288609443 + 5^3Bbb Z = 68 + 5^3Bbb Z$$
$$chi(4) = 4^{5^2} + 5^3Bbb Z = 1125899906842624 + 5^3Bbb Z = 124 + 5^3Bbb Z = 1 + 5^3Bbb Z.$$
(Of course we could have noticed $chi(p1) = 1$ easier.) Notice how they are multiplicative, indeed they are just the set of standard Teichmüller representatives $mu_{p1}(Bbb Z_p)$ modulo $p^n$, and instead of the earlier computation you can write
$$phi_n(bar 1, bar 4, bar2) = chi(1) + 5cdot chi(4) + 5^2cdotchi(2) +5^3Bbb Z = 1+5cdot (1) + 25cdot 57 +5^3Bbb Z = 46 +5^3Bbb Z.$$
Of course that’s all equivalent, but this way you outsource some work into a onceandforallcomputation of the Teichmüller representatives. Also, beware that as soon as one tries to do the same with $Bbb F_q$ instead of $Bbb F_p$, things get more subtle, as one has to put in appropriate $p^i$th roots at appropriate places.
Ad 2), I think it is totally possible to argue this way, although the proofs of that theorem which I have seen (which would be the one of Bourbaki in Commutative Algebra 9, and of Serre in Local Fields) actually go through the Witt vector machinery and thus might exhibit a bit more structure. I just want to point out that, if $q=p^k$, then another description of $Bbb Z_p[mu_{q1}]$ is: the ring of integers (a.k.a valuation ring) of the unique unramified degree $k$ extension of $Bbb Q_p$.