Hello!
You're almost correct. :)
1. You write a letter to your friend and send it.
2. Start the Society Phase and draw a Major Arcana card. You don't need to wait for your friend to receive your letter.
3. Start the Chronicle Phase whenever you're ready for it. You don't need to wait for your friend to have received your letter to do this.
Ideally, you should receive and read a letter before you go back to (1) and write another letter. You may want to save some of your Chronicle Points to react to/explore any topics in your friends letter (but you don't have to).
Basically, as soon as you've sent your letter you continue to your next phase independent of what your friend is doing. The goal is to have received a letter before you write again, to avoid overwhelming one person with more letters than they can handle. :)
I hope that helps clear things up! :)