It appears that there are three camps of stochastic processes:

(1) Old. Analyze bumps on railroad tracks, sound noise, the weather over time, ocean waves, .... So, the expectation is fixed, that is, does not vary. The sample paths are bounded. Might assume that the variance is fixed. If increments are independent and identically distributed, then can assume the distributions are Gaussian (have a Gaussian process).

In that case can do interpolation, extrapolation, smoothing, find power spectra, see what happens to sample paths as you run them through a time-invariant linear filter.

(2) Newer. Can get into Markov processes and maybe also martingales. There is a good text by Cinlar, long at Princeton. His chapter on the Poisson process is especially good. Cinlar does not mention measure theory, but other texts do or really are all about measure theory, that is probability based on measure theory. Can get the measure theory out of Royden, Real Analysis and Rudin, Real and Complex Analysis. Should have a measure theory background in probability, and for that try any or all of Breiman, Neveu, Loeve, Chung. Can get into potential theory. Then stochastic optimal control. May end up with some expensive books from Springer written by Russians. Otherwise I gave some names above.

(3) Queuing Theory. Get to do a lot of work with the Poisson processes with the hope of making applications, e.g., in the sense of operations research.