길 corresponds to a road or way. 거리 means the streets and blocks of a city busy with people and shops and passing cars. Roadway-related words like 건너다, 찾다 (find), 잃다 (be lost) usually go with 길 because 거리 doesn't have a strong sense of a long route. 거리 focuses on the looks and mood of the busy city blocks, so 거리를 건너다 is rarely used (거리를 지나다 = go past a street or block, is more common). It is almost always 길/찻길을 건너다. 큰길 means wide or major thoroughfare, as opposed to 작은 길 (small road/street) or 샛길(byway). 길 is also used widely in the sense of 방법/방책 (a way or method). e.g. 문제를 해결하는 데는 이 길밖에 없어 = This is the only way there is to resolve this issue.