It feels like I submitted my application for asylum in the U.S. so long ago and I still haven't gotten a decision. I'm wondering if I can do anything while I am waiting? I was thinking about taking a college course or getting a job. Also, am I allowed to move somewhere else within the United States?
Both U.S. Citizenship and Immigration Services (USCIS) and the Immigration Courts (EOIR) are extremely backlogged. Thus it can years for them to process or issue decisions in both affirmative asylum applications (the kind you file on your own) and defensive asylum cases (the kind you raise during removal proceedings in immigration court).
You will definitely want to plan your activities during this long wait.
People with pending asylum applications or cases who have been waiting a long time without a decision are allowed to apply for employment authorization. The length of time is currently 150 days (though the Trump Administration tried to change it to 365 days).
In order to apply, you'd need to submit Form I-765, Application for Employment Authorization along with the receipt notice showing the date when your Form I-589, Application for Asylum and Withholding of Removal was received by either the Immigration Court or USCIS and other documents (such as evidence of lawful U.S. entry) showing that you're eligible. See How to Apply for a Work Permit While Awaiting an Asylum Decision for more information.
Since you are legally authorized to remain in the U.S. while your asylum case is pending, you should be able to attend higher education classes as well, though you might not be eligible for certain internship or work study programs in which students in the U.S. on an F-1 student visa can participate.
Also, it will be up to the college or university whether or not to grant you in-state tuition rates or to allow you to take courses for credit.
You can also move to another location within the U.S. while your asylum application is pending, but be sure to notify either USCIS or the immigration court of your change of address as soon as possible. This may also delay the processing of your application or case.
If your case is in immigration court, you will also need to file a Motion to Change Venue if you are moving outside the jurisdiction of that court. Consult with an immigration attorney to help you do this.