You can search on Instagram using keywords to find photos and videos, hashtags, accounts, audio, tags, and places. Tap at the bottom to go to Search & Explore. Tap Search in the search bar at the top. Type who or what you want to search for, then tap the search button in the bottom right of your keyboard.

The only way to see someone's private Instagram profile or stories is to follow their account. Unfortunately, there are no workarounds—you'll need to tap the Follow button on their profile and wait for the person to approve your request. Beware of apps that claim they can get around Instagram privacy settings.