H-Elements are historically and traditionally used and replied on.
P-Elements are the "new-old kid on the block (ahem .. as of 20 years ago)"
H-P Elements are a hybrid approach.... (not all FEA systems support these)
Voxels seems to be the newest trend - Real-Time GPU Powered Solutions.
This really depends on your needs and goals for the type of analysis you are interested in performing (statis, dynamic, contact, collision, meta physics, etc.)
While FEA is fundamentally a great thing, each technology has its benefits and gaps. Edge cases tend to showcase the limitations better (e.g. column deformation is usually a good reference test case).
Higher level mesh resolution vs compute time vs accuracy is a balance... experience with your problems will determine the balance. Typically higher boundary resolution conditions are more desirable (especially in acute geometry - because that is where the stress tends to be more pronounced).
My recommendation - start out coarse, and refine as you learn more from faster analysis to add resolution where necessary to obtain more relevant analysis results.