[noun] + が/を + たくさん + [verb]
たくさんの + [noun]
The noun keeps its own particle, が or を, and たくさん takes none. It works for things you count and things you don't, so no counter is needed.
Directly in front of a noun it needs の: たくさんのひと人, "many people". たくさんのひと人がいます and ひと人がたくさんいます mean the same thing, and both are everyday Japanese.
These go beyond saying how much there is.
Said on its own, もうたくさんです means "that's enough" – you have had as much as you can take, and you want no more.