How can I 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. You can 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 box beside “Add into the cycle on day” checked. 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 switch for Confirmed opt-in.