**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.][1] 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.][3] 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][2]) 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. 


  [1]: http://www.math.jussieu.fr/~leila/SchnepsGT.pdf
  [2]: http://mathoverflow.net/questions/133059/belyis-theorem-for-function-fields
  [3]: https://www.imj-prg.fr/~leila.schneps/farb.pdf