I recently taught a short course on Bayesian Macroeconometrics, and you can find the course notes here.
Below I list some additional resources.
For beginners who want to learn about Bayesian econometrics and computations, the two books that I particularly like are Bayesian Econometrics and Bayesian Econometric Methods. My book also has chapters on Bayesian inference and Markov chain Monte Carlo methods.
If you already know basic Bayesian computations and want to learn more about state space models, Chan and Jeliazkov (2009) would be a good place to start. This paper considers a simple algorithm to estimate linear Gaussian state space models. It illustrates the methods using a dynamic factor model and time-varying parameter vector autoregression.
After linear Gaussian state space models, the next step would be univariate stochastic volatility models. Chan and Hsiao (2014) gives a textbook treatment of a plain vanilla stochastic volatility model as well as two variants. It discusses the auxiliary mixture sampler of Kim, Shepherd and Chib (1998), which is implemented using the precision sampler of Chan and Jeliazkov (2009). See also Chan (2013) for more complex SV models.
The next step is to learn some general algorithms for fitting general nonlinear state space models. It is an active research area and there are many different approaches. In my completely impartial and unbiased opinion, the best place to start is Chan (2017) that considers an accept-reject Metropolis-Hastings algorithm. For two examples using nonlinear state space models for inflation modeling, see Chan, Koop and Potter (2013) and Chan, Koop and Potter (2016).