Here is my problem:
Let $S_g$ be an orientable closed curved surface (like sphere or torus) with genus $g$. With arbitrary $g\geq3$ show that $S_g$ is a regular covering of $S_2$.
I have no idea how to approach this problem since I struggle a lot in writing proofs concerning covering spaces. Any pieces of advice would be greatly appreciated.
Let us deal with standard covering first, then we will see the normal/regular one. A standard trick when dealing with this kind of problems is to use multiplicativity of the Euler characteristic $\chi$. In particular, for a $k$-sheeted cover $E$ of a compact CW-complex $X$ the following formula holds: $$\chi(E)=k\cdot \chi(X)$$ This simple formula (which is not hard to prove, I believe you can find a proof on any standard algebraic topology book) gives you a good starting point when dealing with finding covering spaces of a manifold.
In this case we plug-in $E=S_g$ and $X=S_2$, with the formula $\chi(S_g)=2-2g$ to obtain that the degree of the covering must be $k=g-1$. (you should do these computation to see why we assumed $g\geq 3$, actually $g=2$ is not a big problem :) ).
Now the covering maps are given in the following way. Draw your oriented surface $S_g$ with one of the hole in the centred and the other (which are $g-1$!) evenly spaced around it. The free action is given by rotating around the central hole. Note that it does define a (free) group action of $\Bbb Z/(g-1)\Bbb Z$ on $S_g$ and you should prove that the quotient is really $S_2$ (basically notice that you are identify all the "rays" emanating from the central hole - this gives you intuition on why you get $S_2"- then formally prove that the quotient has all the properties required).
A similar setting is depicted by Hatcher (page 73)
Where the difference is that the quotient is $S_3$ and therefore on each ray we have $2$ holes. You should work out the details of this.
For what concern regularity of the cover let me be a little bit sketchy:every covering action arising as nice group action like this (I'll refer to prop 1.40 page 72 of Hatcher for the precise statement) gives rise to a regular (normal in Hatcher notation) cover.