I'm looking for a probability textbook that would first build intuition using finite sample spaces and uniform probability mass over the sample space in a first part (coin toss, die, Bernoulli Urn, etc.)
Then introduce Kolmogorov axioms and tackle infinite sample spaces in a second part. It's best if there are interesting applications in this second part.
It goes exactly by the scheme you've drawn: first elementary things, then axiomatic theory. This way it is adapted to the sctructure of Probability Theory courses in Moscow University: elementary Probability Theory on the first year of study, followed later by the axiomatic theory, which goes side-by-side with Measure Theory.
The book is quite advanced and not easiest too read (although not so hardcore as Kallenberg; I'd put it on approximately the same level as Kai Lai Chung). This is not bad by itself - I like books which challenge a reader, and Shiryaev, in my opinion, gives a challenge exactly a bit below the level of scaring off an unprepared reader. And it is still very dydactic, a pleasure to read.
All in all, one of my favorite textbooks; my own lecture course is heavily based on it.
I think William Feller's An introduction to Probability theory and applications might be what you are looking for.
