What would we do, if we figured out asteroid mining

Well first, I think we'd mine asteroids.

After that, there are a number of factors that would impact what we'd do with mined materials:

1. In order to mine asteroids in a worthwhile way, we'd likely need to invent autonomous technologies to do so, otherwise we're likely not going to be able to do so effectively/efficiently. That will include the ability to autonomously build autonomous mining/refining/manufacturing facilities.

1a. If we're able to do that, we'll then have the technologies available to efficiently build space-based and planet-based habitats and manufacturing facilities.

2. Assuming we're able to accomplish (1), the obvious route would be to build more mining/refining/manufacturing facilities. This would allow us to create a space-based industrial base. However, it's not clear if such a base can provide materials and finished goods for those (pretty much all of us) living in a gravity well.

3. As such, I think the greatest benefit from learning how to mine asteroids will be in applying the autonomous technologies required on the Earth.

Precious and rare metals. Imagine a rather low price of gold, platinum, and the like which will improve electronics chemistry and manifactoring processes. Maybe, a space lift one day, but we are going into speculation territory there.