길 means a way(most of its senses), road, pathway, and a means of doing something.
거리 has a much narrower meaning as a street or boulevard where cars and people pass by.
So 길 focus on getting to one's destination, while 거리 is just a place, usually in the city, where a lot of people and cars are passing by.
- 서울로 가는 길 => The road to Seoul. (road)
- 여기서 부산까진 먼 길이다 => It's a long way from here to Busan (journey)
- 그 친구를 찾을 길이 없다 => There's no way to find him (way, method)
- 뜻이 있는 곳에 길이 있다 => Where there is a will, there is a way (way, solution)
- 구도자의 길 => The way of a truth (tao) seeker.
- 눈길 => gaze. Sight line. The stare of the eye.
- 손길 => touch, movement of a hand.
- 발길 => path, movement of the feet.
- 거리는 가로등 불빛으로 환했다 => The street was brightly lit by the street lights.
- 토요일 거리엔 행인들이 넘쳤다 => The Saturday street was overflowing with passers-by.
Busy streets are also called 길거리, a compound word made of the two words.
- 골목 안에서만 놀고 길거리엔 나가지 마 => Play staying in the alleyway. Don't go out to the street.