Before launch
On launch day
Lower the DNS TTL a day earlier. Switch DNS at a quiet hour. Check the site from mobile and desktop, from your office and from a mobile network. Check the error log. Make a fresh backup straight afterwards.
The first week
Watch the search console for crawl errors, the access log for 404s, the uptime monitor, and your inbox for messages from people who found something broken. Fix the broken links and add missing redirects as they surface.
After the first month
Review speed tests, update plugins, and read your analytics for what people actually use. Put a recurring reminder in your calendar for renewals and a quarterly restore test.