Which integration to pick, and how to check it works
Pick the integration that matches where your site is built, and install only one. On WordPress, use the plugin: it injects the script for you and adds WooCommerce event tracking. On Shopify, use the Shopify integration so add-to-carts and purchases are recorded without cookies. On a React, Next.js, Vue or Svelte app, use the @eurometrics/tracker npm package, which gives you typed functions for pages, events and goals. For anything else, a hand-written site, a builder that allows custom code, a static page, paste the tracking script tag. Running the plugin and a manual script tag at the same time counts every pageview twice, which is the most common setup mistake.
After installing, verify before you trust any numbers. Open your site in a normal browser window, click through two or three pages, and check that the visits appear in your dashboard. On single-page apps, also check that moving between routes records a new pageview and not only the first load of the app.
Use the REST API when the data needs to live somewhere else: an internal dashboard, a client portal, or a weekly spreadsheet. The API reads visitors, pageviews, sources, geography, events and goals, and it can also manage websites and goals, so an agency can create a site record for each new client from its own tooling. Plan your polling around the limit. Pro plans allow up to 1,000 requests per hour, so fetch a daily summary once and cache it rather than calling the API on every page load of your own dashboard.
Here is a worked example for a WooCommerce shop. Install the WordPress plugin, confirm pageviews arrive, then place a test order and check that the purchase event appears. Create a goal on that event so conversion rates by source show up. Finally, point a webhook at your team chat or ticket system for goal completions and uptime alerts, so the people who need to react hear about it without opening a dashboard.