For an introduction and motivation to TLA+ I recommend Hillel Wayne's talk Everything About Distributed Systems is Terrible [0] and basically anything he has written on his blog about the TLA+.

I think following along with the TLA+ video course is really thorough and digestible once you've decided that you want to learn more [1].

You'll find the TLA+ Hyperbook is a useful reference and guide as you start specifying your own systems [2]. It has a number of examples and projects to work on.

Eventually if you want to go deep after you've decided it's worth it for yourself, get into Specifying Systems the text book and bible of TLA+ [3].

Program Synthesis has a lot of research in it and is starting to show promising results. I'd say that while the research has been going on for a long time it's still rather new to industrial use... for a motivating talk and explanation on what program synthesis is try this one: https://www.youtube.com/watch?v=HnOix9TFy1A

[0] https://www.hillelwayne.com/talks/distributed-systems-tlaplu...

[1] https://lamport.azurewebsites.net/video/videos.html

[2] https://lamport.azurewebsites.net/tla/hyperbook.html

[3] https://lamport.azurewebsites.net/tla/book.html