For the Successful Lineage aspiration, as part of phase 4, you have to mentor a child.
First, you'll need to purchase the Mentoring reward from the Satisfaction rewards. I'm not in game at the moment but I think it was 1000 points.
My child was already a teen at that point so I couldn't use the creativity table so I'll have to edit this post the next time I have a child and tell how that works.
To mentor a child after they've 'grown up', there is a limited number of skills that you can mentor in once you've maxed the skills out. The skills are:
Fitness
Guitar
Handiness
Logic /Chess
Painting
Piano
Violin
Writing
None of the other skills can be mentored.