Learn Russian: Мы переезжаем в новый дом.

The prefix пере- often means moving 'across' a space from one side to another, or making a transfer. Combined with a verb of motion by vehicle, переезжать literally means 'to drive across' or 'to relocate', which is how Russians express moving to a new home or apartment.

I know 'ехать' means to go by vehicle. Why is the verb here 'переезжаем' and not 'переедем'?

When adding a prefix to verbs of motion, Russian uses different verb stems to show if the action is ongoing (imperfective) or a one-time completed result (perfective). For ongoing or repeated trips by vehicle, prefixed verbs use the stem -езжать. So, переезжаем means 'we are moving' right now. Мы переедем is the completed perfective future meaning 'we will move'.

Since 'дом' is a physical location, why don't we say 'в новом доме'?

Because the sentence involves active motion towards a destination. When you move 'to' or 'into' a place, the preposition в takes the accusative case (в новый дом). You only use the prepositional case (в новом доме) to describe being statically located inside the house.

Is the double 'е' in 'переезжаем' pronounced as one long sound or two separate sounds?

It is pronounced as two distinct sounds. You clearly pronounce the end of the prefix пере- and then the start of the root -езжать, so make sure not to merge the two е vowels together.