The ideal $I=(x^2+y^2+z^2,x^2-y^2-z^2+1)$ can also be written $I=(x^2+1/2,y^2+z^2)$, as you have computed yourself.
Now you must study the scheme $S=Spec(k[x,y,z]/I)$.
The trick is to write $k[x,y,z]/I=k[x]/(x^2+1/2)\otimes_k k[y,z]/(y^2+z^2)$, so that the scheme $S$ is the product $S=T\times U$ where $T=Spec(k[x]/(x^2+1/2))$ and $U=Spec( k[y,z]/(y^2+z^2))$.
You will then have to discuss cases , according as each of $-2$ and $-1$ is or not a square in $k$.
For example, if $k$ is algebraically closed then $S$ has four irreducible components.