General method for approaching block/rotating pulley/rotating object equations

Not asking about a specific problem, but general methods. I'm having trouble with problems that usually involve tension, rotating objects, rotating pulleys, and one block free to fall due to gravity. I'm not asking for help with a specific problem, but with this approach in general. It seems like it should be simple, but for whatever reason I always freeze up whenever I try to solve one of these problems, so I'm trying to better understand rotation.

My current line of reasoning is to use force diagrams and then
Since $\tau$ = RF sin ($\theta$) =I$\alpha$,
F =((I$\alpha$)/(R sin($\theta)$))

Since the net force is generally provided by one or more objects allowed to fall
ƩF$_{y}$ (generally equal to m$_{falling object}$g- F$_{tension}$ )=m$_{total}$a + ((I $_{1}$ $\alpha$)/ (R$_{1}$ sin ($\theta)$))
+ ((I $_{2}$ $\alpha$))/ (R$_{2}$ sin ($\theta$) ))... and so on for any of the other I$\alpha$ and ma.
I will then generally plug in a/R for $\alpha$

At this point, I'm generally not sure what to do. Possible ideas are to do a sum of torques equation, or to attempt to solve for Tension in the x direction.

Another thing that confuses me is when to include translational velocity for a rotating object in my equations.

kuruman