Learn Italian: Noi andiamo a Venezia domani. - We are going to Venice tomorrow.

In Italian, you always use the preposition a when talking about going to or being in a specific city, like Venezia. If you were going to a country or a large region instead, you would use the preposition in (for example, in Italia).

Since domani means tomorrow, why is the verb andiamo in the present tense?

Just like in English when we say "we are going", Italian frequently uses the present tense for planned events in the near future. The word domani gives it all the future context you need, so a future tense verb isn't required.

Do I have to keep noi at the beginning of the sentence?

No, it is completely optional. The verb ending in -iamo already tells us the subject is "we". You could simply say Andiamo a Venezia domani. Including noi just adds a little emphasis, like saying "We are going."

I've seen per used for destinations. Could I say per Venezia here?

Not with the verb andare (to go). You must use a for cities when you are going to them. You would only use per (for) with a destination if you were using the verb partire (to leave), as in Partiamo per Venezia (We are leaving for Venice).