In an isothermal process, how can the change in internal energy be 0?
The internal energy of an ideal gas is simply $$ U = \alpha nRT,$$
where $$\alpha = \frac{\text{degrees of freedom}}{2}$$
So, in an isothermal process,
$$
\Delta T = 0 \Longrightarrow \Delta U = \alpha nR\Delta T = 0,$$
and likewise any
$$\left(\frac{\partial U}{\partial P}\right)_T = \left(\frac{\partial U}{\partial V}\right)_T = 0.$$