Lets pretend I have no experience with statistics...
what books, in what order, will get me to the point of being able to understand Bayesian probability fastest.
The folks at LessWrong have put together a couple of resources that can be helpful for the complete beginner. They are also useful if you've only taken a single course in probability and basically just memorized formulas to get through it.
For two more advanced books that cover practical matters in great detail (and require a bit more mathematical maturity) see:
My own opinion is that learning the formula for Bayes' theorem is elementary; it's something that almost always precedes even a first introduction to common discrete random variables. The real "starting point" for seeing the difference in Bayesian methods is when you start needing to compute Maximum Likelihood Estimators (MLE). Then you can directly compare the output of an MLE with the alternative output if you had set up a prior and chased the computation out all the way to a posterior distribution.
There's nothing special about MLEs, of course, but that just happens to be one of the first places where a student is really forced to say, "Gee, I'd really like to have the whole posterior distribution, but it's really hard to compute." Once a student sees the value in that, they are usually ready to start learning about acceptance/rejection, Metropolis-Hastings, and Gibbs sampling, which are the gateway drugs of Bayesian analysis.
Here is a good list. There are lots of material online too, and some excellent q&a right here on stackexchange.
Reference Book: Bayesian Theory by José M. Bernardo and Adrian F. M. Smith
Read: Notes from MIT OCW
Presentation: Tutorial from Stanford
Watch videos on Youtube (about Bayesian Probability)