Here is my own (admittedly, personally biased) list. I do not claim it is better
than anyone else's list, but at least I do know them all very well, having taught undergraduate stochastic processes courses out of each in various decades.
mathematically rigorous, and all are below the measure theoretic level. A full dose of lower-division calculus and perhaps a
beginning course in probability would be helpful background.
All of the authors have clearly used much of the material in
real applications, which I think is important for beginners at the undergraduate
level. Listed in order of publication.
1) Appropriate parts of Feller (vol. 1): A classic book with many surprisingly difficult problems, but heavy on intuitive explanations.
2) Parzen: Somewhat unusual approach and collection of topics, but based on real-world concerns, and with intuitive explanations. (Rigorous enough for many years of use at Stanford.)
3) N.T.J. Bailey: Many examples from biological sciences, but
mathematically rigorous. Includes time-continuous processes.
4) Roe Goodman: Meticulously clear, often intuitive. Some attention to simulation, but not as a substitute for rigorous presentation.
More attention to queueing and other time-continuous processes
than in many books accessible to undergraduates.