At what time can I look up at the sky and be looking in the same direction the earth is moving?

Around 6 am you should be able to look more or less straight up in the direction Earth is traveling, if you account for the axial tilt. Generally from midnight to noon there should be a way to look in the direction Earth is traveling, again if you account for the axial tilt.

To be extremely accurate:

On the day of the equinox (March 21 or September 22), stand on the equator. At exactly "sunrise", look straight up (towards your zenith) and that is where you are going, at roughly 30 km/s (67,500 mph) relative to the Sun.

From other places on Earth (and at other times in the year) you would still be going "head first" at sunrise, but at an angle. The extreme case would be if you are standing at the North pole: point towards the horizon at 90 degrees to the right of the Sun, and that is the direction.

Of course, at that moment (and at any other moment), the Sun is itself orbiting the Galaxy at roughly 200 km/s ( 450,000 mph); compared to that, Earth "puny" orbital speed is negligible. You'd have to find the constellation towards which the Sun appears to move.

Worst: the Galaxy (and the entire Local Group, for that matter) is "falling" towards the Virgo Supercluster at a speed of roughly 400 km/s (almost a million mph). Compared to that, Earth's orbital speed is definitely negligible.

The Virgo Supercluster is called that, because it is seen in the direction of the constellation we call Virgo.

If you can spot the Big Dipper, follow the arc of the bear's tail. You will hit a bright yellow star called Arcturus (in the constellation Bootes). Continue straight south (same distance) until you find an almost-as-bright star called Spica (in the constellation Virgo).

"Follow the arc to Arcturus, then drive a spike to Spica."

In general, that is the direction that the whole system is going to.

Unfortunately (for the demo), the constellation Virgo is presently in the same direction as the Sun (more or less) making it difficult to find.

If you wait a few months, you could find it in the early morning sky (just before dawn). It is easiest to see in late Summer, early Fall.

Since the Earth is round it can only have one point on the surface where the direction of travel is straight up. But if you are are content to go for looking mostly forward...

Looking from above the orbital plane, down at the north pole, the orbit and the rotation are both anticlockwise (counter clockwise). Thus by sketch and inspection, the region going through sunrise always includes the "forward point". Thus when you are going through sunrise, that is as close as it gets for your upwards being forward.

Note that if you watch the sunrise and make a rightangle on the surface where you stand, with one leg pointing at the sun, the other leg points to the "forward point" if you trace it over the surface of the Earth. It will be roughly near the equator somewhere between the two tropics, depending on the time of year.

Sunrise.