Use 'amount of' before singular things you cannot measure.
Use 'quantity of before singular or plural things you can measure.
Use 'number of' before plural things you can measure.
The term 'amount of' is used for things you cannot measure. It usually precedes a singular word.
I undertook an inordinate amount of work.
(work - singular and cannot be measured or counted)
She had a certain amount of respect for the sales team, but she always dissented when they spoke at meetings.
(respect - singular and cannot be measured or counted)
It is not difficult to see where that amount of hate derives.
(hate - singular and cannot be measured or counted)
The term 'quantity of' is used for things that you can measure. It is usually applied to inanimate objects. Some grammar references may advocate that it can only precede a singular word; however, this view is now considered outdated.
I took control of a large quantity of money.
(money - singular and can be measured or counted)
The ship was only carrying a large quantity of mangoes.
(mangoes - plural and can be measured or counted)
('Number of' could also be used in this example. 'Number of' is preferable as it is unlikely to be considered incorrect by some readers.)
The term 'number of' precedes a plural, countable noun. It can be applied to both animate and inanimate objects.
The disease affected a large number of camels in the town.
(camels - plural, animate and can be counted)
The ship was only carrying a large number of mangoes.
(mangoes - plural, inanimate and can be counted)
('Quantity of' could also be used in this example.)