Let $$(x,y,z)=f(\theta,\gamma )=(\sin \varphi\cos\theta,\sin\varphi\sin\theta,\cos \varphi).$$
Therefore, $$\frac{\partial }{\partial \theta}=(-\sin\varphi\sin\theta,\sin\varphi\cos\theta,0)$$ $$\frac{\partial }{\partial \varphi}=(\cos\varphi\cos\theta,\cos\varphi\sin\theta,-\sin\varphi).$$
Q1) How can I compute $\Gamma_{ij}^k$ for $i,j,k\in\{\varphi,\theta\}$. The thing I know is that $$g=\sin^2\varphi\mathrm d \theta^2+\mathrm d \varphi^2$$ and that $$\nabla _{\partial i}\partial _j=\sum_{k\in\{\theta,\varphi\}}\Gamma_{ij}^k\partial _k$$ where $i,j\in \{\theta,\varphi\}$. But to be honnest, I really don't know how to apply those formula to get Christoffel symbol. I recall that for example $\partial _\theta=\frac{\partial }{\partial \theta}$.
Q2) An other thing I don't understand, it's with the definition of $\partial _\varphi$ and $\partial_\theta$ above, how will I compute for example $\frac{\partial g}{\partial \theta}$ ? where $g$ is a function (for example, $g(\theta,\varphi)=\theta^2\varphi$) . How will I apply this ?
$$ g_{\theta\theta}=\sin^2\varphi ~~~~~~ g_{\varphi\varphi}=1 ~~~~~~other=0 $$ Then,only $\partial_\varphi g_{\theta\theta}=\sin 2\varphi$ , other $\partial g =0$. By $\Gamma_{ij}^k=\frac{1}{2}g^{k\mu}(\partial _i g_{\mu j}+\partial _j g_{\mu i}-\partial _\mu g_{ij})$, have $$ \Gamma^\theta_{\varphi\varphi}=0 \\ \Gamma^\theta_{\theta\varphi}=\Gamma^\theta_{\varphi\theta}=\frac{\sin 2\varphi}{2\sin^2\varphi} \\ \Gamma^\varphi_{\theta\theta}=\frac{\sin 2\varphi}{2} \\ \Gamma^\varphi_{\theta\varphi}=\Gamma^\varphi_{\varphi\theta}=0 $$
When you compute the $\Gamma$ , it is just complex, don't be afraid. Riemannian geometry always be complex . Just my opinion ,and I am beginner too.