A few little things definitely worth remembering across the topics:
SHM) if you set up your SHM scenario but get an acceleration proportional to displacement or theta rather than -displacement or -theta, then you are simply taking your direction in the opposite of the actual, this can be easily fixed.
Circular motion) conditions for there to be a circular motion,
String: T>0N (otherwise we are dealing with a projectile and M2 projectile motion can be used)
Cylinder: R>0N (same as above)
Rigid rod: v>0 at top (the rod can't become slack)
Outside of sphere: R>0N (projectile)
Variable forces) remember the integral of f'(x)/f(x) = ln(f(x)) and you can save some time, recall all of your C4 integration and separation of variables.
Velocity = dx/dt
Acceleration = dv/dt = vdv/dx (a result from product rule)
DON'T forget your constants
Hookes law) recall that T= (modulus of elasticity)(extension)/(natural length), this can also be written as F=kx where k is stiffness however this is rare (I imagine) and just thing of k as (lambda)/(L)
Don't forget EPE= (lambda)(extension)^2 / 2(natural length)
Or EPE= 1/2 kx^2
Rigid bodies) remember if the two rods are identical, and the system is symmetrical in the horizontal plane (e.g two identical rods jointed at top and with same angle to floor/ horizontal) then you only have to worry about horizontal forces on pin joint. However in most cases (unless if they felt particularly nice) it is not a symmetrical system.
Remember it is best to find an expression for cos(theta) and sin(theta) early on to help with calculations.
Impulse and momentum) I=mv-mu
If you have a ball hitting a horizontal smooth surface then the parallel component of velocity remains constant whereas vertical component is dictated by coefficient of restitution (remember to get the directions right). And for two balls colliding then velocity perpendicular to line of centres is the same, but parallel components found using coefficient of restitution and momentum.
Don't forget impulse triangles, they can be fiddly to set up and you must therefor eve careful with the angles used.
Good luck