In my Main Timeline, I've got 3 symbols (circles with an icon).
First Symbol's Timeline starts to play automatically. The Timeline of Symbol 2 doesn't play automatically. I don't think I have added something to start playback for Symbol 1, but I can't figure out, why Symbol 1 runs, but Symbol 2 does not.
A Hype file can help us advise on the structure or if you are hitting problems.
There are a few different ways to control symbol timeline playback, I suspect you'd just want to add a Symbol Action (which is in the properties timeline when the symbol element is selected). Please see this documentation:
If you want to run more than one symbol timelines with the same action, you can hit the "+" button on the action and add another Start Timeline… event.
There's a few cases where Hype will automatically add symbol actions to play the main timeline on your scene's main timeline (like when creating a symbol), but in most other cases Hype cannot intuit the most likely behavior and you will need to manually add them. Hype never adds symbol actions for a symbol's non-main-timelines.