Can you use Invisibility, grab some gold, and have it then be invisible? Or does it only work on items on your person when the spell is cast?
Unfortunately, the rule about this is a little ambiguous.
Anything the target is wearing or carrying is invisible as long as it is on the target’s person.
This has 2 interpretations that are, as far as I can tell, equally valid.
Anything the target is wearing or carrying at the time you cast the spell is invisible as long as it is on the target’s person.
Anything the target is wearing or carrying at any point throughout the spell's duration is invisible as long as it is on the target’s person.
Luckily, someone has asked Jeremy Crawford, official source of rules interpretation for D&D 5e, and he says that it is the first one.
Not "really", but it depends.
According to designer Jeremy Crawford you cannot make stuff you pick up invisible, but you can hide it under your invisible clothes (or bags I guess, or inside your fist maybe?). The item is still visible if you put it back down (or hold out).
And remember that dropping your own items makes them visible too because "anything the target is wearing or carrying is invisible as long as it is on the target’s person."
I would also like to add that this is one of those cases where the rule is only a little bit muddy, but both interpretations are "balanced" and logical (not silly) and can potentially create interesting situations or solve/create different problems. I mean seeing an object flying on its "own" can lead to various stories and so can things disappearing. And you can even use both. Maybe one mage's version of invisibility works slightly different to another. You can even make it a (very) minor plot point. As long as you keep it somewhat balanced (in your player's eyes at least, they can get cranky at times).