Akumulator i istrošena baterija idu na odvojeno sakupljanje.

Because the subject is made of two nouns joined by i (and):

• akumulator
• istrošena baterija

Together they form a plural subject, so the verb must be plural:

• idu = they go

So:

• Akumulator ide = The accumulator goes
• Akumulator i baterija idu = The accumulator and the battery go

Serbian often uses the singular in signs, rules, and instructions to name a category of item in a general way.

So here:

• akumulator = an accumulator / car battery as a type
• istrošena baterija = a used battery as a type

This is similar to how English instructions can say something like Plastic bottle goes in the blue bin, even though it really means plastic bottles in general.

A plural version is also possible:

• Akumulatori i istrošene baterije idu na odvojeno sakupljanje.

That would sound more explicitly plural, but the singular version is normal in instructional language.

In Serbian, these are related but not identical words.

• akumulator usually means a rechargeable storage battery, especially a car battery
• baterija usually means an ordinary battery, like an AA or AAA battery

So the sentence is distinguishing two kinds of waste:

• akumulator = larger rechargeable battery, often automotive
• istrošena baterija = spent/dead battery

This distinction is very common in Serbian recycling and disposal language.

Istrošena literally means something like worn out, used up, or spent.

With baterija, it means a battery that has been used and no longer has charge, so in natural English this is often:

• spent battery
• used battery
• dead battery

Grammar-wise, istrošena is the feminine singular form because it agrees with baterija, which is feminine singular.

Because adjectives in Serbian must agree with the noun they describe in gender, number, and case.

Here:

• baterija is feminine singular
• so the adjective must also be feminine singular
• therefore: istrošena

Compare:

• istrošen akumulator = a worn-out accumulator
  because akumulator is masculine
• istrošena baterija = a worn-out battery
  because baterija is feminine

It is the accusative.

The preposition na can take different cases, but here it expresses movement/direction toward a destination or purpose:

• idu na... = go to / go for...

So:

• sakupljanje is neuter singular
• in the accusative singular, neuter nouns look the same as nominative
• odvojeno agrees with sakupljanje, so it is also neuter singular

That is why you get:

• na odvojeno sakupljanje

Because odvojeno describes sakupljanje, and sakupljanje is a neuter singular noun.

Agreement works like this:

• masculine singular: odvojeni / odvojen
• feminine singular: odvojena
• neuter singular: odvojeno

Since the noun is:

• sakupljanje = neuter singular

the adjective must be:

• odvojeno

So odvojeno sakupljanje means separate collection.

Sakupljanje means collecting or collection.

It is a verbal noun formed from the verb sakupljati / sakupiti, meaning to collect or to gather.

So:

• sakupljati = to collect
• sakupljanje = collecting / collection

In environmental or recycling language, odvojeno sakupljanje is a standard phrase meaning:

• separate collection
• separate waste collection
• selective collection

Because na is idiomatic in many expressions involving processes, events, services, or destinations of use/disposal.

Serbian often says:

• ići na pregled = go for an examination
• ići na reciklažu = go for recycling
• ići na obradu = go for processing

So ići na odvojeno sakupljanje means something like:

• to go for separate collection
• to be sent to separate collection

Using u here would not be the usual phrasing.

It sounds like instructional / recycling-sign language, not everyday casual speech.

A native speaker would understand it perfectly, but in ordinary conversation you might also hear things like:

• Akumulator i istrošena baterija odlažu se odvojeno.
• Akumulatori i istrošene baterije skupljaju se odvojeno.
• Akumulator i istrošena baterija treba da idu na odvojeno sakupljanje.

So your sentence is very natural in the context of:

• recycling instructions
• disposal rules
• public signage
• packaging labels

Because Serbian does not have articles like English a/an and the.

So:

• akumulator can mean an accumulator, the accumulator, or accumulator(s) in a generic sense, depending on context
• baterija works the same way

The meaning is understood from the situation. In this sentence, the nouns are used generically, so the sense is:

• accumulators and used batteries as categories of waste

Yes. The verb ići literally means to go, but in this kind of context it often has a broader practical meaning:

• belong in
• should be placed in
• are sent to
• go to the stream/category of

So idu na odvojeno sakupljanje is not mainly about the items moving by themselves. It means that they are supposed to be disposed of through separate collection.

Here is a simple breakdown:

• Akumulator — accumulator, usually a car battery
• i — and
• istrošena — worn out, spent, used up
• baterija — battery
• idu — go
• na — to / for
• odvojeno — separate
• sakupljanje — collection

So literally it is something like:

• Accumulator and spent battery go to separate collection

A more natural English rendering would be:

• Car batteries and used batteries go to separate collection
• Car batteries and spent batteries should be collected separately
• Car batteries and used batteries belong in separate collection