Just things I've sponged up over the years:
Looking at the rocket as a whole, a dominant load on the entire structure is compression. The load starts as a static 1g weight supported by the structure at the base, then multiplies as it accelerates during launch. Aerodynamic drag works against the body moving through the air, then adds buffeting and sonic shock as the craft travels fast enough.
Thin sheets of metal do not behave nicely when they are compressed on their edges, making this particularly tricky. Various strategies are employed, such as long, thick reinforcing members running up the sides, or maintaining the pressure inside the integral fuel tanks to prevent the wrinkling/buckling. Example of the former, Titan (Gemini program) and the former, Redstone (Mercury program).
Looking at components in more detail, and the story changes depending on the component. Fuel tanks often carry cryogenic liquids, meaning they must maintain their shape through dramatic shifts in thermal expansion while holding the internal pressure. Engines, of course, experience tremendous heat, pressure, and vibration. Joints between stages must withstand all interface loads, yet separate reliably at the right moment.
There are also loads caused, ironically, by the very lightness of the rocket itself, or more accurately, by its flexibility. Made from such thin flexible materials, a rocket can experience many kinds of whole-body oscillations in lateral and axial directions. One of these is called "pogo" and just like the toy, the rocket's thrust makes it shorter, then it springs back longer, then back, and this oscillation doesn't damp down until the capsule on top it getting several extra g's of +/- acceleration. I read that the Titan II (Gemini program) had that problem.
There are also "pre-launch" loads, which occur before the rocket gets to the pad. If the machine is transported laying horizontally on a truck/barge/train/etc then you wouldn't want a component crushed by the weight.
As SpaceX is now proving, there will soon be "post-launch" loads to consider, as their rockets come down and land themselves. With the intent to re-use the rocket, this matters now.
STF