In grammar, this is a battle between what we call "countable nouns" and "uncountable nouns". Funnily enough, a "countable noun" is anything there is a number of that you can count. An "uncountable noun" is generally a singular thing, which of course can only be counted once.
Whether to use "amount" or "number", "much" or "many" depends entirely on whether the noun is countable or not. I just can't help but get jitters when I see or hear them used incorrectly.
If you want to get it right, the basest thing you have to do is decide whether the thing you're talking about is countable or not.
A cake is an uncountable noun. There is only one of it. When discussing an uncountable noun, you use "amount" or "much".
I couldn't handle the amount of cake on my plate. As soon as I was finished, I ran to the bathroom to lurch. (Or if you're Australian, "toilet".)
"How much cake would you like, dear?"
Skittles are countable nouns. There are more than one of them in the packet. When discussing a countable noun, you use "number" or "many". (This is where I often see people make the mistake, so take note.)
The number of Skittles in the packet had slowly dwindled until there were none. (Note that you don't say "amount" because the Skittles are countable.)