Do I need to tell Google I changed hosts?

No, not for a straightforward move to new hosting where the domain and every URL stay the same. Google's index is built around URLs and their content, not the server behind them. If both are identical before and after the move, there is nothing about the site that has actually changed from Google's point of view — it will simply notice, in its own time, that responses are coming from a different IP address, which is not something it treats as significant.

When you genuinely do need to tell Google

The situation is different if the migration also involves a domain change — moving from olddomain.com to newdomain.com. In that case, use Search Console's change of address tool, which exists specifically to tell Google that a site has permanently relocated and that ranking signal should transfer accordingly. See moving a website to a new domain for the full process, including the redirects that need to be in place before you use that tool.

Switching between http and https, or between www and non-www, is also technically a URL change even though it can feel like a minor detail — each is a different address as far as a search engine is concerned, and both benefit from proper redirects even if no formal "change of address" declaration is needed.

What is worth doing instead, for a plain host move

• Confirm your Search Console property is still verified after the move — verification can occasionally rely on a file or DNS record that needs recreating on the new host.
• Check the sitemap is still accessible at its usual address.
• Watch for a short-lived dip in crawl stats immediately after the move, which is normal and should settle within days rather than persisting.

What actually affects rankings during a host change

Not the move itself, but downtime, broken pages, or URLs that stop resolving — none of which are caused by changing server, only by a migration that was rushed or done out of order. See will changing host hurt my search rankings for the full picture.