How can I have multiple users on this program if my whole family wants to work at their own pace?
You can use the same windows (I presume you're using it) account. Use the Google Chrome browser. Under the settings (the three horizontal bars on the top right), go to Settings. There you can add different users. Several windows with different users can be open at the same time. Create one user for every family member, and when they want to use Duolingo, they open their own Chrome instance, and log into Duolingo. If there is already one Chrome open, check the icon on the very top left. If you left-click on it, you will be presented with all Chrome users present on the system. Clicking on one opens a new window.
One great thing about this is that you can even log in as this user on another computer, and have all your settings and passwords transferred there :) But be very cautious with this feature, as chrome will save your passwords in CLEAR TEXT on every machine you ever logged in (you can remove your profile, but people usually forget).
OR you can just use different browsers :) One uses Chrome, the other Firefox. The loser gets Explorer :P