How do I determine the resultant permittivity of two different dielectric materials in a cylinder capacitor?
You can reduce this problem to a series combination of two separate cylinder capacitors.

One with inner radius \$r_0\$, outer radius \$r_1\$ and permitivity \$\epsilon_{r1}\$ and
the other with inner radius \$r_1\$, outer radius \$r_2\$ and permittivity \$\epsilon_{r2}\$.

Since they are in series, charge (per length) will be equal for both capacitors.