Players argue about pick shape. Shape matters least of the three.
Gauge is a mechanical decision
A thin pick bends as it crosses the strings, which spreads the attack out over time and softens it. That is why strumming with a heavy pick sounds percussive and slightly brutal, and why picking single notes with a thin one sounds vague.
It is not a matter of skill level, despite how picks are often sold. It is a matter of what you are playing.
Material is a tonal decision
Celluloid: warm, rounded, traditional, wears quickly. Nylon: softer attack, noticeably quieter, grippy when damp. Delrin (acetal): brighter and more defined, slick, extremely hard-wearing. Ultem: hardest and brightest, an audible click on the attack.
Put the same player with the same guitar behind four picks of identical gauge in those four materials and the difference is obvious in a second.
Then shape, and mostly for control
A small jazz shape gives you less material to twist between the fingers, so the tip stays where you aimed it. A sharp tip gives a more defined attack than a rounded one.
These are real but small next to gauge and material, which is roughly the opposite of how picks are usually marketed.
What to specify on an order
Material by name, gauge in millimetres, shape, and whether you want a textured grip. Four short answers.
If you are buying for a group with mixed players, buy two gauges rather than compromising on one. It costs almost nothing extra and it is the difference between a pick people use and a pick people put in a drawer.