You need the IP addresses of all the equivalent services on the new host; You also need to know their policy on changing those IPs in the future, which they might do without telling you. (why? because usually they host the DNS, so they move your site to another server, and just change the DNS themselves. But when you’re in Cloudflare, they no longer can manage your DNS).
You could find the IPs on their service by either their control panel (if it shows them), or by running dig or nslookup against their DNS servers which they told you to use - and your domain and www.domain addresses.
THEN, you can put this/these IP/s in your HOSTS file on your computer, to check that the website works correctly in the new location. Verify in your browser developer tools that indeed you’re connecting to the NEW IP and not the old one (it might be cached). Best method is to make a special page that exists only on the NEW host, and browse to it. If you can see it - it means you’re pointing to the new server. If you don’t - check what you did wrong.
AFTER you’ve verified that the site works when all your site’s hostnames point to the new host’s IPs, it is time to make the changes you did locally - global. Remove your HOSTS records and restart your browser. Then go to Cloudflare’s DNS, and change all the relevant hosts (usually domain and www.domain - but you may have others - and you should have checked them ALL prior to the move) - to the IPs that you tested in your HOSTS file.
If the site is protected by Cloudflare (orange icon), then the move of traffic should be immediate. You should be able to immediately access domain/the-the-page-that-you-created-that-exists-only-on-the-new-host.