1. Monitor https://hackerone.com, https://bugcrowd.com and Twitter for announcements of new programs.
2. When looking for bugs in sites with existing programs like Facebook your best chance is when they announce a new feature or product. This includes acquisitions (Facebook paid out over $100,000 for bugs when they added the Oculus websites to their program).
All bug bounty programs have rules that outline what parts of their site/product you can test and what kinds of bugs they are looking for (here's Facebook's https://www.facebook.com/whitehat/). As long as you follow the rules you won't have any legal problems.
In terms of how to get started, I definitely suggest monitoring the various bug bounty sites to see what's new and if a bounty's scope has expanded.
There's also a bunch of guides, tutorials, and tools listed on Bugcrowd's Forum: https://forum.bugcrowd.com/c/security-research