Where can I find information about monthly seasonal change in a given area? I have searched Google for almost an hour with no result. I'm looking for something like, "Indonesia, January to August is rain season, September to December is hot season".

I can't use the old information in books about seasonal change that now seem to have shifted and I can't find any updated information anywhere.
You can try climatemps.com for easy access and plots. There they have a map you can click on or a list by countries.
More data and analysis tools are available at the KNMI Climate Explorer, a more reliable and scientifically sound source.

This is one of the outputs for Jakarta from jakarta.climatemps.com