The waxing crescent moon appears to the east of the sun and move farther east each day. Does this mean it rises earlier each day or later?

later

Later

Explanation:

The Moon is only natural satellite of Earth and goes in an elliptical orbit around it. This revolution period is of about 29.5 days. That means it covers 360° in 29.5 days which means it covers around 12.20° in one day.

Also, during this revolution the Earth also moves 1° in its own orbit.

Thus, the Moon needs to cover 13.20° extra every day to rise. Moon covers 1° in 4 minutes so to cover 13.20° it will need extra 52.8 minutes.

Hence, each day the Moon will rise later by 52.8 minutes.