Poslužićemo hranu i piće.

Because the verb form -ćemo already shows that the subject is we. Serbian often leaves out subject pronouns when the verb ending makes the subject clear.

You could add mi if you want emphasis or contrast:

Mi ćemo poslužiti hranu i piće.

It is the future tense: specifically future I in Serbian grammar.

The basic verb is poslužiti = to serve, and ćemo is the future auxiliary meaning we will. So poslužićemo means we will serve.

In Serbian, with many verbs whose infinitive ends in -ti, the future auxiliary can attach directly to the verb.

So:

poslužiti + ćemo → poslužićemo

A separate version is also possible in a sentence such as:

Mi ćemo poslužiti hranu i piće.

When ćemo is written separately, it usually appears in the second position of the sentence.

The dictionary form is poslužiti.

That is the infinitive form, the one you would find in a dictionary. In this sentence it appears in a future-tense form: poslužićemo.

Poslužiti is a perfective verb.

That means it presents the action as a completed whole: we will serve as one finished act. That fits this sentence well.

An imperfective verb such as posluživati would suggest something more repeated, habitual, or ongoing, like we will be serving or we usually serve depending on context.

Because hranu is the accusative singular form of hrana.

Here food is the direct object of the verb, and direct objects often go in the accusative case in Serbian.

• hrana = nominative, dictionary form
• hranu = accusative, used here

Because piće is a neuter noun, and for many neuter nouns the nominative and accusative singular are identical.

So even though piće is also a direct object here, its form does not change:

• piće = nominative
• piće = accusative

Yes. They are singular, but they are being used as mass nouns.

So hrana i piće means food and drink in a general sense, not necessarily one single food item and one single drink. This is very natural in Serbian.

If you wanted to talk about separate dishes and beverages, you might use words like jela i pića.

Yes. Serbian word order is fairly flexible.

This sentence is natural as written:

Poslužićemo hranu i piće.

But you can also say:

Mi ćemo poslužiti hranu i piće.

or

Hranu i piće ćemo poslužiti.

The meaning stays basically the same, but the emphasis can shift depending on which part comes first.

• ž sounds like the s in measure or vision
• ć is a soft sound, somewhat between English t and ch

Very rough approximations:

• poslužićemo ≈ po-sloo-zhee-cheh-mo
• piće ≈ pee-cheh

But keep in mind that Serbian ć is usually softer than normal English ch.

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

So Serbian simply says:

hranu i piće

and the context tells you whether English should be food and drink, the food and drinks, or something similar.