How to setup a contact journey and add them in specific mailing group after a delay?
I would 


Use webform(Drupal) or Caldera(Wordpress) for signup
Setup webform(Drupal) or Caldera(Wordpress) to create activity of type XXX as completed status with todays date on submission.
Setup Schedule reminder for Activity of type XXX

a. 10 days after activity date

b. 20 days after activity date
Setup 2 smart group 

a. Contacts having activity with activity datetime for past 10 days of type XXX

b Contact having activity with type XXX and not in group 4.a


Cheers

Pradeep