Getting started

Hobby astronomers wanting to get started with astrostatistics need the following:
  1. A computer. Any reasonably recent machine will do. Clearly, the more powerful your computer, the more computationally demanding procedures you are able to run. 
  2. Astronomical data. You will have to find a data set online containing data on astronomical phenomena of interest to you. Some pointers are given in the Data section of this website.
  3. Statistical software. Popular open source packages include R and Python. For now, this websites has a strong focus on R, see the Software section.
You could start exploring some of the Projects presented on this website, and try to repeat, vary, extend and improve the analyses.

Book

A good book with useful examples is Modern statistical methods for astronomy - with R applications, by Feigelson and Babu, 2012. 

Games

A fun way to conduct data analyses is to take part in online contests, such as those organised by Kaggle. Every now and then they have a contest involving astronomical data. Examples from the past are: Mapping dark matter, Observing dark worlds, and The galaxy challenge.  

More

The website Astrostatistics and Astroinformatics Portal (ASIAP) contains a wealth of information on the subject, mainly for professionals but the website is certainly also informative for hobbyists.