How do I host my Google App Engine App as a site at my domain, and transfer form submissions to a database?

Background: My app is a single-page HTML form that has the fields "Name", "Email", "Phone", then a drop-down box with 3 options, and lastly a "please contact me" checkbox. I created it with create-react-app, then made a production build, then added an app.yaml file using the configuration instructions from Google Cloud Platform. I then pushed the build folder and app.yaml file to my remote GitHub repo, and then used the Google Cloud Shell to clone that GitHub repo into Google App Engine.

I deployed my app, and google gave me the link to view my app in my browser, the myproject.ue.r.appspot.com link.

Now I want to do two things:

1. Host my app (the form) at the domain I already own through google domains.

2. When someone visits my domain and fills out the form, then clicks submit, I want to save all the entered fields as a new record into a database, so that I can contact each person that submitted the form. I can see that Google Cloud Platform has Filestore and Storage, would this be where I should start?

I've been reading through the guides on Google Cloud Platform, but because of my limited knowledge, I'm not sure if I'm on the right track.

I don't necessarily need exact answers, I would greatly appreciate it even if someone could just direct me to the correct guides to read.

Your help and patience is greatly appreciated