Do meteorites streak across the sky in the same direction? Once while camping with my family as a child we all watched a meteor shower in the summer (in the month of August).

All the meteorites streaked across the sky in the same direction.

Obviously, this was the result of the Earth passing thru a group of meteors but I've always wondered.

Do meteorites always streak in the same direction? While I know it's not likely to be exactly the same direction, but generally speaking. Would one say meteorites always travel for example: east to west?

I thought this was a plausible fact since the Earth is travelling around the Sun in a constant direction, and additionally it's always rotating in the same direction.

While meteorites can travel in all directions. Does the Earth's gravity pull them straight down towards the surface resulting in the streak to be always in the same direction.
No, meteorites of the same shower look as coming from the same point in the sky. They are usually named according to the point they seen to come from, e.g. the leonids, or the perseids. The point, they seem to originate is a result of the relative velocity of the meteorites to the earth.

Meteor showers are thought to be remnants of comets, hence the similar orbits of the meteorites within a shower.

Here a more complete list of meteor showers.