The Moon does not make its own light. It always has one half lit by the Sun; what changes is how much of that lit half faces us.
The 29.5-day cycle
As the Moon orbits the Earth, the angle between the Sun, Earth and Moon changes. At Amavasya (new moon) the Moon sits roughly between us and the Sun, so its lit side faces away and we see nothing. About 14.8 days later, at Purnima (full moon), the Moon is on the opposite side of the Earth from the Sun and we see the whole lit face.
In between we see a crescent (less than half lit), a half moon (the quarter phases) and a gibbous moon (more than half lit). The growing half of the cycle is Shukla Paksha; the shrinking half is Krishna Paksha.
Why the moon rises later each day
The Moon moves about 12–13° eastward against the stars every day, so it rises on average about 50 minutes later each night. That is why the full moon rises around sunset, while the moon on Karva Chauth (four days after full moon) rises 2–3 hours after sunset.
Is the shape the same across India?
Yes. Everyone on Earth sees the same phase at the same moment. What changes from city to city is the time it rises and sets, and the tilt of the crescent.