How to do it:
Open Telegram.
Tap the Search icon (magnifying glass) at the top of your chat list.
In the search bar, type the person's exact Telegram username (it usually starts with an @, but you don't need to type the @ in the search bar itself).
Telegram will display matching profiles under "Global Search Results" (or similar).
Tap on the correct person's profile to open a chat.
You can then send them a message. To formally "add to contacts" (so they appear in your Contacts list), tap their profile picture/name at the top of the chat, then select "Add to Contacts". You'll usually have the option to set a name for them.
Why it's good: You don't need their phone number, and they won't automatically see yours (if you've set your phone number privacy to "Nobody").
2. Adding by Phone Number (If you have their number)
If you already have someone's phone number telegram number database saved in your phone's address book, Telegram can automatically suggest them or you can add them manually.
Automatic Discovery (if Contact Sync is ON):
If you have "Sync Contacts" enabled in Settings > Privacy and Security > Contacts, Telegram will automatically check your phone's address book. If a number there is registered on Telegram, that person will appear in your Telegram contacts list. You'll also receive a "Contact Joined Telegram" notification (if enabled).
Note: For privacy, many users disable "Sync Contacts."
Manual Addition (Recommended if Contact Sync is OFF):
Open Telegram.
Tap the New Message icon (a pencil/paper icon, usually in the bottom-right on Android or top-right on iOS).
Enter the person's First Name and Last Name (optional).