If you want to challenge one of your friends to a game of chess on Chess.com, it's as easy as a few clicks. See here for how to add someone as a friend! This article references the website. See here for the Android app and here for the iOS app.
One way to challenge a friend is to hover over 'Play' in the left menu, and click 'Play'
Then, in the right hand menu, click on 'Play a Friend'
From here you will be able to search for your friend's username, or create a challenge link to send to someone via a messenger app or email, or you can choose from a list of your friends who are online.
Once you have chosen your friend, you will be given another menu where you can select the time setting, and type of game. If you and your friend are very far apart in rating, you may want to try an Odds Game. See here for more info on Odds Games.
Remember that your friend must be online in order to accept the challenge! Otherwise the challenge will expire after a few minutes.
If you are sending a request for a Daily game, it doesn't matter whether your opponent is online, offline, or in a game. The request will wait for them to accept it.
Another way you can challenge a friend is by simply going to their profile, and clicking on the challenge icon:
Or, click on their name anywhere you see it, even in chat or your game history:
One last way you can challenge your friend is by using a shortcut url: chess.com/play/USERNAME
Replace 'username' with the username of the member you want to challenge, then put the url in your address bar, and it will take you directly to a challenge to that member!
For the above options you do not need to be friends with the member to challenge them!
Mobile
To challenge one of your friends on your mobile device, first tap on 'New Game' on your homescreen to get to the new game menu. Then tap on 'Play a Friend'
In the next menu, you can choose which of your friends to challenge. Or if you know the username of someone you want to challenge, type it in the username field!
You can also Invite Friends from this screen!
Once you've picked your opponent, you will be taken back to the custom game screen, and you will now see your friend's name as your opponent:
When you have all the settings as you like them, and are ready, tap 'Play' and your friend will receive the notification of your challenge! When they accept, the game will start!