Learn Croatian: Posjetit ćemo prijatelja sljedeći tjedan.

When an infinitive verb ending in -ti comes right before the future tense helper word (like ćemo), it drops its final -i. If you flipped the order, it would keep the -i: Mi ćemo posjetiti.

Where is the word for 'we' in this sentence?

The 'we' is built into the future helper word ćemo (we will). You could add the pronoun mi at the start (Mi ćemo posjetiti...), but Croatian usually drops it because ćemo already makes it perfectly clear who is doing the action.

Why does prijatelj (friend) end in -a here as prijatelja?

It is the direct object of the sentence (the person being visited), so it needs the accusative case. For masculine nouns that are alive, like a friend, the accusative case adds an -a to the end.

Sljedeći tjedan means 'next week', but shouldn't it have a special ending for the accusative case too?

It actually is in the accusative case! Time phrases like this use the accusative, but because tjedan (week) is a masculine noun that is not alive (inanimate), its accusative form looks exactly the same as its basic dictionary form.