I have gone through the settings but haven't figured out a way to do the following.

I would like for new members to be able to post immediately after registration but have their post approved before it appears. This should be the case for a few days, let's say 3. After that, they should be normal members without needing approval.

Any tips on that?

It's per group setting. Go there, open Content tab and find Moderation section:

Thank you! It seems so obvious in retrospect. My mistake was that while going through the settings, I didn't click the require approval button so the next line didn't appear and it seemed like a permanent setting.