As Lawrence said, selecting a default browser and selecting a search engine are two entirely different and independent things.
To select the search engine you want to use, go to Settings>Safari>Search Engine. Your choices are Google, Yahoo, Bing, Yandex, DuckDuckGo and Ecosia. You do not need to download any additional browsers.
To choose a different default browser, first, you need to download the browser app from the App Store. Then, you need to go to Settings and scroll down until you find the new browser app. Tap on it. Tap on Default Browser App and make your selection. The options you see will depend on what apps you have installed. If Safari is your default browser, you won't see any options to change the default browser under Settings>Safari.