Summarize:

Tap the Facebook app icon, which resembles a white "f" on a dark-blue background. Doing so will open your News Feed. If you aren't logged into Facebook, enter your email address (or phone number) and password before continuing. It's in either the bottom-right corner of the screen (iPhone) or the top-right corner of the screen (Android). This prompts a pop-out menu. This option is near the top of the pop-out menu. Tapping it opens the Groups page. It's the dark-blue bar at the top of the screen with the "Search Groups" text in it. If you want to look through groups that you belong to, stay on the Groups page instead. You can see all of the groups you own or participate in here. Type in a word or a phrase that corresponds to the type of group you'd like to find. As you type, results will appear below the search field. If you're looking for a specific group, type in the group's name instead. Tap the name of a group that you'd like to take a look at. This will allow you to see the group's cover photo and posts if it's a public group.  If the group is closed, you won't be able to view the posts. You cannot find a secret group without being invited by a group member. This blue button is near the top of the page. Doing so will send a request to the group administrator. If you're accepted into the group, you'll be able to revisit it by selecting the Groups tab from the ☰ menu.
Open Facebook. Tap ☰. Tap Groups. Tap the search bar. Enter a search term. Select a group. Tap Join Group if you want to join.