Learn Dutch: We beginnen vandaag met ons project. - We are starting our project today.

Both ons and onze mean "our". You use ons with singular het words (like het project) and onze with de words or any plural word. Because it is het project, we must use ons.

The English translation says "starting our project", so why is the word met (with) in the Dutch sentence?

In Dutch, the verb beginnen (to start or begin) is almost always followed by met when you mention what you are starting. You are literally saying "We start today with our project."

Why is vandaag (today) placed in the middle of the sentence? Could I put it at the end?

It is much better to put time words like vandaag right after the verb in Dutch. Putting time words at the very end of a sentence, like we often do in English, sounds unnatural to Dutch speakers.

How does beginnen translate to "are starting"? Shouldn't there be a word for "are"?

Dutch does not use a separate "are doing" or "-ing" form for things happening right now. The simple present tense word beginnen covers both "we start" and "we are starting".