Your public profile is a page at /@your-handle that anyone can open. You set the handle in Settings > Account.
What's on your profile
Your name, @handle, and avatar
A bio, up to 500 characters
Up to 6 social links
Your follower and following counts, plus the lists behind them
Wishes from your Public wishlists, and nothing else
Change your handle
Go to Settings > Account.
Edit your handle. It's checked as you type, so you'll know straight away if it's taken.
Save. Your profile link becomes the new
/@handle.
A handle has to be 3 to 50 characters, unique, and not a reserved word. Your bio lives on the same page.
Good to know
Two settings pages sound alike and do different jobs. Settings > Account controls the public profile: your handle and bio. Settings > Profile sets your display name, photo, language, and country. Your name and photo do show up publicly.
There's no switch that makes your whole profile private. Visibility works per wishlist instead, so set anything you'd rather keep back to Invite only or Anyone with the link. Those never appear on your profile, whoever is looking. See "How is my public profile different from sharing a single wishlist?".
Profile pages ask search engines not to list them, but that's a request rather than a guarantee. And anyone holding your /@handle link can open the page regardless.