On the home course page you will have a list of all assignments, all due dates are in central time. The first type of assignment your teacher can assign you in Quest is a web assignment. You can recognize a web assignment by the 
icon. To view the assignment click the link next to the icon.
The questions themselves will be viewable in an embedded PDF, you can choose to open this PDF in its own tab or download it. The assignment will have instructions at the top of the page followed by the questions you have to answer.
When you are ready to submit your answer, select it in the "My Answers" menu on the right and then click on "submit".
Once you have submitted answers you will either get a 
(for a correct answer) or "Started" (for an incorrect answer) under status, the tries column will tell you how many attempts you have left on each question and points will tell you how many points you have currently earned on that question.
Once you have completed the assignment your grade will appear under the my answers section. It will also appear next to the homework on your course home page.
