A round up of the second video:
First they show the cost (money) and benefit (fans, money and experience, in this order at the bottom) of the various actions you can do. They were shown in this order: Promotion (free to cheap), audition (free), live (expensive) and festival (free). It doesn't look like they changed a lot from im@s 2. The main differences i'v seen so far are the bonus for a certain appeal during auditions (as seen in the first video), and that a festival rival's burst doesn't prevent you from using your appeals anymore.
Next up, lessons: The text doesn't reveal more than the screenshots. It doesn't explain the breath bar for example, it just says you need to pay attention. For the dance lesson the buttons to press and the rhythm must be correct. The visual lesson seems to have changed completely, and honestly i'm happy about it, they were a pain in im@s 2.
In the rank up festival you need to win twice. Additionally, the rivals burst's are associated with a skill. Looks like Ai drastically lowers the scroe you get during appeals while her burst is active. And since Miki bursts right at the beginning of the second song it seems you can carry over the burst gauge. After 2 victories the leader idol gets a rank up.
A rank up improves various abilities (it's not mentioned what kind of abilities) and a special communication that shows new sides of the idol.
The next video will be about producer rank ups an the half yearly all stars live with all 13 members.
I hope this post is of any interest to anyone :p