How to send an instant message after someone signs up via the landing page?

For the landing page subscriptions, use an autoresponder set to day 0.

In order to send an instant message to contacts who sign up via the landing page, you need to have an autoresponder created first. Your can do it under Tools>>Autoresponders>>Create new. You can also do it under Dashboard>>Create>>Create autoresponder. Create your autoresponder message for day 0, and select the desired list.

In the Landing page settings scroll down to Subscription settings and make sure you have the toggle switched to blue next to “Add contacts to an autoresponder cycle”. If you ask to confirm the subscription your subscribers will need to confirm first in order to get the autoresponder. To disable double opt-in for the landing page, turn off the toggle next to Send an opt-in confirmation email.