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. (I will look up a precise reference)
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 $p$. (This should also be in SGA1, I will look it up). In characteristic $0$, you can compare the etale fundamental group to the profinite completion of the topological fundamental group. 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 $$ 1\to \widehat{F_2}\to \pi_1^{et}(\mathbb{P}_{\mathbb{F}_q}^1\setminus\{0,1,\infty\})\to \widehat{\mathbb{Z}}\to 1. $$ The action of the Galois group of $\mathbb{F}_q$ on the geometric fundamental group is determined essentially by the action of the Galois group on the points of $\mathbb{P}^1/\overline{\mathbb{F}_q}$.
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 $\operatorname{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\cong \widehat{Z}$.