First, there are a few different methods developed to solve this kind of problem, but it's highly dependent on your background knowledge (calculus), and experience with these kinds of problems. This approach might be over-detailed for some mechanics problems, but this approach is fairly general, so should usually work.
- Identify any clues in the problem as to how the objects in the system to move. In this case, the problem tells you the bead drops. Because the problem is symmetric, you expect the bead to drop straight down rather than move sideways. The problem says the rings roll over a "sufficiently rough surface". This identifies that the rings don't slide, so rotation of the rings has to go hand-in-hand with their horizontal motion.
- Think through how you expect the objects to move. As the bead falls, the rings should roll outward, and the rings should probably slow down the fall of the bead.
- Draw the free body diagram for each object in the system. I've drawn one for one of the rings for you. The other ring should have the same diagram. You'd also need to draw one for the bead. Notice that I've also divided the one of the forces into the x and y components. This will be useful later.
- Label the forces with convenient variables. I've already done this on the diagram.
- Choose variables for the motion each object. For the ring, you'd probably choose an angle of rotation $\theta$ and horizontal position $x$. You don't need to bother with vertical position, since that's constant.
- Write the equations of motion for each variable, for each object. The ring will have two equations, one of them $F_x=ma_x$, for horizontal movement, and the other $\tau=I\alpha$, where $I$ is the moment of inertia for a ring, $\alpha$ the angular acceleration, and $\tau$ is the torque or moment of force. You'll have to calculate $\tau$ in terms of the forces of course.
- Use constraints to eliminate variables. Because the ring is constrained from slipping, you can write $\tau$ in terms of $a_x$ and eliminate one of those two variables. From this point on, it's just a game of eliminating unknown variables, and reducing the number of relevant equations until you arrive at the acceleration of the bead in terms of known forces.
Once that one acceleration is known, then you can calculate any quantity you need in the problem.