Does the noun 'garbage' go with 'it' or 'them'?

"garbage" is an uncountable noun and is treated as singular.

It should be "it" :) "I picked up the garbage and threw *it* into the rubbish bin"

I would say though that rubbish bin is the British term and we wouldn't say 'garbage'. It's slightly awkward mixing American and British English in the sentence like that. Nothing wrong with it, just a little unusual.

Is your native language Polish by chance? If my memory serves me right the word “garbage” is plural in Polish, just like “scissors” or “pants” in English.

I would suggest, I picked up the rubbish and threw it into the bin. You're kind of mixing British and American words there.

It sounds right

Singular, but garbage is an Americanism and rubbish bin is Commonwealth English (most of us would omit rubbish and just call it a bin, especially if you already clarified the fact that it’s rubbish you’re chucking)

It would only be “them” if the garbage you were talking about was a person or persons.

Garbage is singular, so “it”

"Garbage", "trash", and "rubbish" are all uncountable, so you say "it". "Them" is for plurals. And as someone else pointed out, British people usually favor "rubbish" and "bin", while Americans usually favor "garbage" or "trash" and "can". So a Brit would put rubbish in the rubbish bin, and an American would put garbage in the garbage can, even though it's the same object and they're cleaning together. 🙂 (Also, I'm not sure if this is regional, but it seems like in my part of the USA, a trash can is usually smaller and indoors, and a garbage can is usually bigger and outdoors.)

It

You’d use “it” but you shouldn’t mix garbage and rubbish. Garbage is American and rubbish is British

Everyone is answering your question fine so I’ll give some other alternatives. As an American I would say trash, and I would just say I threw it “away.” So you could say: I picked up the trash and threw it away. I picked up all the trash and threw it away. I picked up pieces of trash and threw them away. I picked up trash and threw it all away. I picked up pieces of trash and threw it/them all away. If you want to say where exactly you threw it away, then I’d say trash can. We do also say garbage, just not as often. I might say it if I want to emphasize it in a funny way. “Look at all this GARBAGE!!!!!”

Grammatically, "it" is correct, but many Americans will use "them" when referring to multiple things at once (ie: A handful of broken toys rather than a garbage bag). I would assume this is the same in other places, but I've had international friends tease me for it before, and I've yet to figure out who was the weird one in that situation

It's a collective noun so it's treated as singular.  So "it".  Treat it the same way you would treat the name of a substance like water, or sand, or wood.  

The rule is that uncountable things use the singular.  Rubbish / garbish is uncountable so "it".