I understand that buckling is considered for long slender elements under compressive loads. However, I was wondering if there are any rules of thumb for basic mechanical design to help determine whether an element can be considered "long and slender". Which in turn would determine whether a failure due to buckling analysis should be taken into consideration for a particular element.

If the length of the plate is L and the thickness is delta, then Delta<<L, usually it depends on the whether you are using sheet metal or rod, or what kind of boundary conditions you are having for the sheet metal or rod. For example on the top surface of an airplane wing when the wing deflects upwards, the most prominent type of failure is considered to be buckling/ crippling. for every part in a machine there are suspected places where buckling failure might occur. You can ask here in this forum if you are designing such a part. From my experience, if you are using simply supported on all 4 sides, and delta is less than 15L then I always check for buckling failure. If you are having free boundary conditions on two sides and simply supported on the sides where the force in applied. Then delta less than 10L you should check for buckling failure. If you keep these in mind you can mix and match for other boundary conditions. If the boundary conditions are fixed then you know for sure delta can be much less than 15L or 10L.