Chicken Soup

If there are a majority of noodles

Brocho before eating this: Borei Minei Mezonos

Brocho after eating the proper shiur of this: Al Hamichya

If there are no noodles or vegetables

Brocho before eating this: Shehakol Niyeh Bedvoro

In most cases a proper shiur is not consumed, and therefore, no Brocho Achrona is required. However, in the rare case of consuming a proper shiur of this say: Borei Nefashos

If it has larger pieces of celery or carrot

Brocho before eating the celery or carrot: Borei Pri Ha'adama

Brocho before eating the soup: Shehakol Niyeh Bedvoro

Brocho after eating the proper shiur of this: Borei Nefashos

If it has a small amount of noodles, kneidlach, or soup nuts

Brocho before eating the soup: Shehakol Niyeh Bedvoro

Brocho before eating the noodles [...]: Borei Minei Mezonos

Brocho after eating the proper shiur of this [if a proper shiur of Mezonos is eaten]: Al Hamichya

Source: The Laws of B'rachos (page 380)