How do I search a Telegram channel for a specific user's messages?

It can be next to impossible to manually search a Telegram channel for one user's messages, so you will want to use the Search tool. From within a channel, tap on the title of the channel at the top center to reveal some tools. Tap on the Search tool.

Now tap on the person icon. A list of channel members will come up, you can either tap on the user whose messages you want to find, or you can manually enter their user name. Once you search you can navigate through the messages with the up and down arrows.