Email one: deliver, at once
Send what you promised in the first email, straight away. Say in one or two lines what the subscriber will get from you from now on and how often. Keep to one link: the thing they asked for. Anything else competes with it.
Email two: your most useful idea
A few days later, send the single most useful idea, story or example you have. Not a roundup and not a sales pitch: one thing that helps. This is the email that decides whether your name gets opened or ignored from here on.
Email three: one next step
Then make one clear invitation: a free tool, a short call, a product. One, not three. Say who it is for and who should skip it. A subscriber who is not ready yet will still be on the list next month; a subscriber offered five choices often takes none.
The basics that keep you on the right side
Email only people who asked to hear from you, make it easy to unsubscribe in every message, and use an honest sender name and subject line. The UK Information Commissioner's Office guidance on direct marketing by email explains consent and the right to opt out in plain language, and is a sound standard to work to wherever your readers are.
Measure replies and clicks
Open rates have become unreliable, because some mail apps load emails automatically. Judge the sequence by clicks on the one link in each email, by replies, and by how many subscribers take the next step. Ask a question in email one and read every answer: it is the fastest research you will ever do.