Im currently starting my master studies in mathematics and have found PDEs fascination, but more on the applied side rather than the theoretical.
Im wondering if you should study SDE in order to understand ODE/PDE and how they relate? Are there practical applications of SDEs outside of finance?
You don't need to study sde/spde in order to understand deterministic ode/pde. However, stochastic equations are interesting in their own right and you should certainly look into them if you find them to be of interest (Oksendal has some good books). Yes, they have applications outside of finance (e.g., lots of applications in physics and some applications in mathematical biology). As an aside, Martin Hairer was awarded the Fields medal in 2014 for his work on spde (e.g., the KPZ equation, which has applications in physics, and regularity structures).