[muraiki]

Doing Bayesian Data Analysis, Second Edition: A Tutorial with R, JAGS, and Stan. It is very approachable and also has lots of practice problems. It's not a math-heavy book at all.

Edit: I wouldn't recommend Probabilistic Programming and Bayesian Methods for Hackers. When I tried using it, I felt that too much was glossed over. The book that I recommend excels at conveying a strong intuition for how these various techniques work.

[saosebastiao]

I'd second this. DBDA made things click for me where a handful of other books failed.

[j_s]

Doing Bayesian Data Analysis https://amzn.com/dp/B00PYZ2VR6/ ($65).

Free is hard to beat, especially for someone just testing the waters.