Is clearing my browser history enough?
It's a good first step since some websites you Browse *do* install/download spyware or other dangerous files onto your device. But as it can be difficult to catch every bad file that tries to make its way onto your device when you Browse the Internet, or ever be 100% sure your Browser is not tracking you and relaying information about you to a large database without your knowledge, take some advanced steps to better protect yourself. Technology and bad actors are only going to increase how much information they squeeze out of us all, so take steps to secure yourself as soon as possible.
How can I monitor my kids' search history?
While there are some apps and computer software that might be able to help you monitor Browser History / Search History of other people in your home, some of the best things you can do on this subject is to have intentional conversations with your kids about what they may see online, listen to and talk with them regularly on what are acceptable and unacceptable actions when Browsing the Internet, encourage them to maintain certain values, and make good decisions for themselves on actions they take online/on Browsing the Internet. Part of these conversations can be legality of online content, and consequences of bad decisions when Browsing the Internet and Social Media.
Is DuckDuckGo the best browser for safety?
Remember, there is a difference between a "Browser", which is a program/app you have on your device that you use to surf the Internet; and a "Search Engine", which is a website on the Internet that you visit by way of a Browser program/app which is where you enter a search term and run a search for something online. DuckDuckGo has a "Browser" program/app and it is also a "Search Engine" website. That having been said, it appears utilizing DuckDuckGo's services may be safer than other, more-popular Browsers and Search Engines, yes.