I'm having trouble with countable and uncountable nouns. For example, I've heard people say fruits, vegetables, meats, etc., but many grammar books say these nouns are uncountable. True or false? Why?
It depends on the usage:

fruit e.g. can be either an uncountable and a countable noun:

When you mean "fruit" to be a singular noun, e.g. the apple is the fruit of an apple-tree, then it also has a plural: these two apples are the fruits of that apple-tree out there.

When fruit is used as a collective noun, uncountable, then it doesn't have a special plural form:
Would you like some fruit?

It might be the same with meat, but I'm not sure whether "vegetable" can be used as a collective noun the same way?!
Some types of meats and vegetables may be used as collective nouns, but not the word vegetable itself. Squash, ham, okra, steak, pizza, hamburger (as in a generic term for ground meat), cabbage, garlic, and lettuce are some examples.

Now that I think about it, I guess you could use vegetable as a collective noun. For example, if you see some unidentified matter and suspect it to have come from a vegetable, you can simply call the stuff 'vegetable'. "What's that gooey stuff?" "I don't know, but I think it's vegetable."