How to show a function is primitive recursive by induction?
Wikipedia gives such an example. You haven't stated the definition of basic functions or basic operations, so I will assume the Wikipedia definition.
The successor function $S(x) = x+1$ is a basic function, and so is assigned 0. Projection functions such as $P_1^1(x) = x$, $P_2^3(x,y,z) = y$ and $P_3^3(x,y,z) = z$ are also assigned 0. The constant function $z(x) = 0$ is also assigned 0.
The function $g(x,y,z) = S(P_2^3(x,y,z)) = y + 1$ is a composition of two functions assigned 0, and so is assigned 1.
Using these functions, we can define a function $h(x,y)$ by primitive recursion: $h(0,y) = P_1^1(y) = y$, and $h(S(x),y) = g(x,h(x,y),y) = h(x,y) + 1$. This function is assigned 2, and you can check that $g(x,y) = x + y$.
The function $r(x,y,z) = h(P_2^3(x,y,z),P_3^3(x,y,z)) = g(y,z) = y + z$ is assigned 3.
We can define a function $k(x,y)$ by primitive recursion: $k(0,y) = z(y) = 0$, and $k(S(x),y) = r(x,k(x,y),y) = k(x,y)+y$. This function is assigned 4, and you can check that $k(x,y) = x\cdot y$.