Probability by A.N. Shiryaev
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.