Typically speaking, the path to becoming a cave diver goes something like this:
Advanced certification + total of 25 dives.
Cavern course - minimum 4 dives, minimum 2 days.
Intro course - minimum 4 dives, minimum 2 days.
Advanced Nitrox/Deco Procedures - minimum 6 dives (7 with some agencies), minimum 3 days. Can sometimes be included in full cave with agency approval.
Full Cave - minimum 8 dives, minimum 4 days.
There are also two other agencies with a different program (NAUI and GUE) that looks something like this:
75 dives pre-req + either intro to tech or fundamentals
Cave 1 - typically 5 days
Mandatory Experience period - typically go off and do 25 cave dives at the cave 1 level
Decompression Procedures
Cave 2 - typically 5 days
Although the first ("traditional") approach seems to be the quickest route, most cave instructors worth their salt will not run a person all the way straight through to full cave. Cavern/Intro are a common stopping point to then let the student go practice and building experience before coming back for full cave. I would probably run away from a cave instructor that promised to take you straight through from Advanced Diver to full Cave without requiring you to go build some experience on your own in between steps.
How much practice, experience, and time between steps depends on the student and instructor; I have a student that finished Intro with me last year, and wants to take Full Cave, but I am making him go do more dives first. On the other hand, I'm running a full cave course later this month for some students that took intro with me this past spring.