During a forced play, batter hits a ground ball to the first base. The fielder at first base catches the ball then tags the runner while still on the base and then tags the base. Is this a double-play, or is only the batter-runner out and the runner safe?
Your description is correct and that's the way the defense needs to do it to achieve the double play.
If the base were tagged first, it would remove the force and the runner would have the right to remain (meaning only one out).
By 5.09(b)(6) A runner is out if....
(6) He or the next base is tagged before he touches the next base, after he has been forced to advance by reason of the batter becoming a runner.
So tagging the runner puts him out. Standing on the incorrect base affords no protection. As long as the force is in play, he needs to reach the next base to avoid out by tagging.
Once done, the same rule applies to the batter (now become a runner) and he is out when first (the "next" base) is tagged before he arrives.