Being able to see the Moon is dependant on various factors like sunset timing etc. Considering all these factors, which country or area of the world sees the new moon first? I am asking about the new moon.
The duration of one lunation (the period between one new moon to the next one) isn't neither constant as the Moon rotates around the Earth and it around the Sun (changes between 29.272 and 29.833 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit), nor integer divisible by 24 hours or one Earth's rotation around its axis. So this position of the Moon on the skies where the next new moon as the first of its lunar phases will be first observable constantly changes.
Phases of the Moon, as seen looking southward from the Northern Hemisphere. The Southern Hemisphere will see each phase
rotated through 180°. (Source: Wikipedia on Lunar phase)
Saying it differently, by the time the Moon completes one lunation (or it's synodic period) and starts the next one, it won't be positioned exactly above the same Earth's longitude (East to West) as the one it started at.