**The general exact sequence:**

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 $\operatorname{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\cong \widehat{Z}$.

**Partial description of the action:**

At the moment, I can not write down an easy description of the action. Simply translating the definition of the étale fundamental group, 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 $C'\to\mathbb{P}^1_{\overline{\mathbb{F}_q}}\setminus\{0,1,\infty\}$.
The action (and hence the extension) are then given by a homomorphism $\operatorname{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\to\operatorname{Out}(\widehat{F_2})$.

A more precise description of the action can be obtained using Grothendieck-Teichmüller theory (which attempts describing the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $\widehat{F_2}$) and specialization of the étale fundamental group. For a survey on Grothendieck-Teichmüller theory can be found e.g. in this survey paper of Leila Schneps. In §3.1 of this article, you can find the description of Ihara's morphism which describes the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $\widehat{F_2}$:
$$
\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{Aut}(\widehat{F_2}):
\sigma\mapsto \left\{x\mapsto x^{\chi(\sigma)},y\mapsto f_\sigma^{-1}y^{\chi(\sigma)}f_\sigma\right\}
$$
where $x$, $y$ are chosen generators for $F_2$, $\chi:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \widehat{\mathbb{Z}}^\times$ is the cyclotomic character and $f_\sigma$ is the pro-loop given by composing the straight path from $0$ to $1$ with its image under $\sigma$.
The specialization of the étale fundamental group should now imply that the action of $\operatorname{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)$ is given by choosing a Frobenius element and using the following composition
$$
\operatorname{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)\to \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{Aut}(\widehat{F_2}).
$$
This yields a well-defined extension since the Frobenius elements are conjugate, and it incidentally explains how the Grothendieck-Teichmüller theory for $\mathbb{Q}$, $\mathbb{Q}_p$ and $\mathbb{F}_p$ are related.
I still do not know how to get a more explicit description of the extension; what we see from the above is that the action of Frobenius on the first loop is by raising the first loop to the $p$-th power. The action on the second loop is more complicated, because I do not know of explicit formulas for $f_\sigma$.
Something along the lines of studying Grothendieck-Teichmüller theory locally is discussed in section 5 of these notes on open problems in Grothendieck-Teichmüller theory. Maybe some expert on Grothendieck-Teichmüller theory knows how to describe the action in the function field case? Or maybe this is complicated, after all the Frobenius elements are dense?

A final note: 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.