I know this info is available online, but it's my question of the day, so here goes. ;-)
Note> You can set it up twitter to use web or wifi if you have a smartphone; in that case, you may just want to use the appropriate Twitter app for your phone: Iphone; or Android (ton of choices).
On your menu bar, choose> Settings
On the Settings page> Choose Mobile
Use Twitter for text messaging!
1. Choose your country from dropdown.
2. Input your phone no. (change it here if you need to)
>Let others find me by my phone number: That is up to you, but think carefully about that.
Note the cheatsheet for Text Messaging on Twitter; you can always get back to this info via this screen, if needed.
3. Verify your phone> Click start (make sure your phone is on!) (Twitter will then connect with your phone; you must text twitter back in order to complete the process.)
4. Verifying your phone - linking it to your twitter account
The shortcode for the U.S. is 40404; other codes can be found here
Once you receive a message from twitter, you will text back to your shortcode (in my case 40404) the message GO
Once you do that it's done, but you do want to take a look at privacy, security, and CONVENIENCE settings.
Let others find me by my phone number: if you are trying to stay private, you may not want to turn this on.
Text message notifications:
Twitter will send alerts to your phone, depending on your settings. You can limit it to particular times (I would not want to receive alerts in the middle of the night), or particular types of messages. You can limit to only those you allow to send to send alerts via twitter (perhaps, your family, dearest friends, or your boss, if you tweet for work); to direct messages, as well as mentions and replies.
So, you'll want to think about how many times you want twitter to alert your phone and for what. If you do not want twitter to alert you for any reason, make sure all of the notification messages are UNCHECKED.