How do I delete my Google search history, and can I set it to auto-delete going forward?

Go to myactivity.google.com, that's where all your Google activity lives, not just browser history. You can delete by date range, or hit Delete > 'All time' to wipe everything. Better move: click 'Auto-delete' and set it to remove activity older than 3 months automatically, so you never do it manually again. Note this is separate from your browser history, clearing Chrome (Ctrl+Shift+Del) only clears the local browser, while My Activity is tied to your Google account across every device. Do both for a proper clean. You can also pause 'Web & App Activity' entirely if you don't want Google saving it at all.

myactivity.google.com is the important one, that's your account-level history across every device. Clearing Chrome alone leaves all of it intact. The auto-delete (3 or 18 months) is the set-and-forget option.

If you want it to simply stop tracking, pause 'Web & App Activity' in your Google account's Data & Privacy settings. Search still works, Google just doesn't log it to your account anymore.