Switching templates on an existing squarespace site without breaking it

Whatever you do, keep the page URLs the same. Squarespace lets you edit a page's slug easily and it's tempting to tidy them during a redesign, but changed URLs lose whatever ranking they had. If any do change, add 301 redirects in the URL Mappings panel under the developer tools settings. And screenshot every page before you start, much easier to spot what went missing.

Which version are you on? On 7.1 there's nothing to switch, every site uses the same underlying template and you change the look in Site Styles. So you can't break anything that way, just restyle. On 7.0 you can install a second template and preview it while the old one stays live, which is the safe way to do it. Your pages and content come across, but anything template-specific might not (index pages, some gallery layouts, sidebar blocks, custom CSS that targeted the old template's classes). Click through every page in preview before you set it live, and copy your custom CSS somewhere first.