Probabilistic Programming
I’ve been doing a bit of probabilistic modelling in STAN recently and have used JAGS for a long time. Probabilistic modelling (as embodied in probabilistic graphical models  PGMs  and Bayesian statistics and implemented in probabilistic programming languages and libraries like STAN) is a way to model some phenomenon that incorporates various sources of randomness, and the dependence between components of that model. The models used tend to be more sparse and more informative than would be generated by a neuralnetwork based model for the same phenomenon (IMO). However, even for modestly sized models or models that encode hierarchical dependencies between variables, you have to use approximation techniques (MarkovChain Monte Carlo  MCMC, or variational Bayes) during model fitting.
Here are some resources for getting started with MCMC models (and their variants).
Tools

 JAGS was the first MCMC tool I used. At the time it provided a crossplatform BUGSlike syntax. You can call it from R (see this blogpost for details of Rspecific tools for working with JAGS).

if you’ve got no missing data, and all your variables are continuous, you should start with STAN. It uses Hamiltonian MCMC which, although individual steps may be slower than an equivalent JAGS step, STAN typically works out faster than JAGS because you typically require fewer samples during your runs.
There are some case studies, user’s guides etc at the website and there are good tools for integrating with R and python.
STAN models are compiled to C++ before running, so there is a bit of overhead before your sampling steps kick off.

 this is a pythonic tool for doing Bayesian modelling. It is independent of JAGS / BUGS / STAN. I have relatively little experience with PyMC3, but it does have some good resources: the tutorials section is brilliant and the book by Osvaldo Martin provides a good introduction.
There are dozens of other tools for probabilistic programming
Tutorials
The PyMC3 tutorials are really good.
I haven’t found any really good JAGS / STAN tutorials.
Courses
Matthew Heiner and Herbie Lee have a course on MCMC and techniques for Bayesian statistics / probabilistic modelling at Coursera, which I can highly recommend. You’ll need to be able to manipulate probability distributions first though.:
Daphne Koller does a threepart course on Probabilistic graphical models through Coursera (it used Octave / Matlab when I did it, which I didn’t care for) but, at the time (~ 2013?), I found the courses quite eyeopening.
Books
Statistical Rethinking  Richard McElreath
This book is remarkably well written. It’s tied to the R ecosystem but it should be possible to translate the models over to other languages (indeed others have converted the exercises into Python, Julia and other R DSLs: https://xcelab.net/rm/statisticalrethinking). The later chapters use STAN and cover hierarchical models, generalised linear models etc.
Note that his lecture course is available on youtube
Bayesian Analysis with Python  Osvaldo Martin
 Although I’d recommend working through the PyMC3 tutorials instead, this book was pretty good. You may be able to get it for free through the Packt book of the day thing packt
There’s a range of machine learning books that cover probabilistic modelling. Three with a strong Bayesian / PGM bias are the Bishop, Barber and Koller books.
Blogs & Other notes
 Darren Wilkinson at Newcastle Uni has a blog that covers MCMC and also some other stuff I’m into at the moment: functional programming and scala. The Rainier package combines all three of these threads and this blog has included a few posts on rainer recently.