What is the best way to reduce the app space that I am using?

Use the management my billing account page to list your apps and sort them by app space used.

Then look at the apps using the most space and see if there are tables that you can purge out (delete) old records.  You may want to export those records to excel before deleting.

I'm considering registering for the service but according to the Premier plan, I only get 20MB of application space.  That's not a lot of data.  Am I understanding that correctly?

No, you get a lot more space than that.  The limit may be 10 MB per user. but you should put in a support ticket to clarify that.