You don't need - or want - a short code from Twilio: those are very expensive (often many thousands of dollars). Instead you only need a long code, which is equivalent to a phone number. These cost about $1USD/month to get from Twilio.
To get started, you must first set up your own account on Twilio. Sign up at http://www.twilio.com. Within Twilio's system, you will:
Create your own account.
Provide a payment mechanism (usually a credit card).
Purchase at least one incoming phone number from them. You should get one in the same area code as your regular phone numbers, but this number cannot be the same as your regular phone number and must be acquired from Twilio.
From the main Twilio dashboard, click on the "#" sign to see your active numbers.
Click on + to buy a new number.
Search for and select an appropriate number to purchase.
Configure that phone number, within Twilio's website, to look like this. Note especially the SMS URL which is required by Twilio. The URL here does nothing and does not record your messages, but does act as a placeholder so that Twilio can know it's notified someone.