If you aren't comfortable with coding, the easiest way to trigger a newsletter popup to appear when a button is clicked is by installing a third-party app from our Shopify App Store. An example of such app is the Email Popup & Sign Up App here. This app looks to allow merchants to control when a popup should appear, as well as its frequency, which is a useful feature to have. Just a note that apps like this one are made by third-party developers outside of Shopify. So if you need more information about each respective app’s functionality, I recommend getting in touch with the app developers as they'd be best equipped to answer app specific questions.
Are you using a third-party app or integration already, to collect customer email addresses for your newsletter? If you are, you can also double check within your app's settings, website, or you can also reach out to the app's support team to see if the app has a functionality that allows for the popup to show up on trigger. As an example, Klaviyo has a code tutorial here on how to add a popup that displays on a click of a button.
Once you have your popup trigger sorted, you can also start to think about how to best leverage your customer newsletter list by optimizing your email marketing. Email marketing remains one of the best channels for maintaining and strengthening customer relationships over time. It also plays a crucial role in encouraging customers to return to your store and continue purchasing, so it's an incredibly effective tool for driving sales for your store. If you need a guide on email marketing best practices, we have a great one on our Shopify blog here that you can take a look at.