Networking without feeling pushy (update)

I sent 9 messages on a professional network and got two replies. This is my first time dealing with any of this. I want advice, not a job directly. What did you ask?

Follow up with a thank-you and a short update. One of my chats led to a referral a month later.

Thanks for writing this out. I went through something close to it last year, and it settled down once I kept a simple log of every call and form.