Calculate the angular velocity of the earth in its orbit around the sun and about its axis.

A delightful problem !

I'm pretty sure that what we need here is the speeds, not the velocities,
and that's the way I'm going to do it.

Regular speed is  (distance covered) divided by (time to cover the distance) .

Angular speed is very much the same.
It's
                 (angle turned) divided by (time to turn the angle) .

Earth's orbit around the sun:

Once per year.
Roughly 360° in 365 days almost exactly 1° per day.
Let's see what it is more accurately:

   (360°) / (365.25636 days) = 0.985609° per day.



Earth's rotation on its axis:

Once per "day".
Roughly 360° in 24 hours almost exactly 15° per hour.

This one is slightly trickier to do more accurately, because a day is
not necessarily 24 hours. It depends on what you call 1 day. 

-- If you say the day is the period of time between when the sun is
highest in the sky, then that averages out to 24 hours in the course
of a year.

-- If you say that the day is the period of time it takes for a star
to reach the same point in the sky tomorrow night, then that's

                   23 hours, 56 minutes, 4.09 seconds  .

Using this to calculate the angular speed of rotation, you get

                 (360°) / (23h 56m 4.09s) = 15.041° per hour