For any scheme $X$ over a field $k$, there is a sequence $$ 1\to \pi_1^{et}(\overline{X})\to \pi_1^{et}(X)\to\operatorname{Gal}(\overline{k}/k)\to 1. $$ Here $\overline{X}=X\times_k\overline{k}$ in slight abuse of notation, and I am a bit sloppy about the base points here. This sequence can be found in SGA1, Exposé IX, Thm 6.1.
Since $\mathbb{P}^1\setminus\{0,1,\infty\}$ is defined over $\operatorname{Spec}\mathbb{Z}$, you can compare the geometric part of the fundamental group in characteristic $0$ to characteristic $p$. I think this is an application of the specialization theory of the etale fundamental group in SGA1. In characteristic $0$, you can compare the etale fundamental group to the profinite completion of the topological fundamental group, SGA1, Exposé XII, Corollary 5.2. The result is that the prime-to-p part of the fundamental group is (independent of the characteristic) the group $\widehat{F_2}$, i.e. the pro-prime-to-p completion of the free group on $2$ generators.
In the case where the base field has at least $3$ points, there is a $k$-rational point of $\mathbb{P}^1\setminus\{0,1,\infty\}$, in which case the sequence splits. The result is a split extension $$ 1\to \widehat{F_2}\to \pi_1^{et}(\mathbb{P}_{\mathbb{F}_q}^1\setminus\{0,1,\infty\})\to \widehat{\mathbb{Z}}\to 1. $$ This leaves the action to be identified.
The same then applies for $\mathbb{P}^1$ with $n$ points removed. Topologically, the fundamental group is a free group on $n-1$ generators, so the geometric fundamental group (resp. its prime-to-p part) is the pro-prime-to p-completion of the free group on $n-1$ generators. The étale fundamental is an extension of this by $\operatorname{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\cong \widehat{Z}$.
At the moment, I can not write down an easy description of the action. What follows from the description of the étale fundamental group is that the action is essentially described as follows: take an étale covering $C\to\mathbb{P}^1_{\overline{\mathbb{F}_q}}\setminus\{0,1,\infty\}$, this is defined over $\mathbb{F}_{q^n}$, and pullback along Frobenius gives another covering $C'\to\mathbb{P}^1_{\overline{\mathbb{F}_q}}\setminus\{0,1,\infty\}$. To get a more precise description of the action, you can do the same thing that is done in characteristic $0$: the function field analogue of Belyi's theorem (about which you can read here) shows that the above is in fact an action on the algebraic curves over $\overline{\mathbb{F}_q}$ because all these are covers of $\mathbb{P}^1$ ramified at at most 3 points.
Maybe some expert on Grothendieck-Teichmüller theory knows how to describe the action in the function field case? The function field case should be a lot easier, and worked out somewhere, as the morphism $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{Out}(\widehat{F_2})$ is replaced by a much simpler map $\widehat{\mathbb{Z}}\to\operatorname{Out}(\widehat{F_2})$.
Sorry, but this is about as far as I can go with what I know...