Learn Mandarin Grammar: V1 + 着 + (O)+ V2 + (O) vs. 一边 ... 一边:

Verb1 + 着 + (Object)+ Verb2 + (Object) is used to explain two actions happen in the same time to a person. It's essentially the same translations with一边...一边 but the emphases are slightly different.

Examples:  

1a. He is driving while listening to music. (It emphasizes the situation in which the person is listening to music)

他听着音乐开车。

1b. He is listening to music and driving at the same time (He is listening to music while driving, or He is driving while listening to music)

他一边听音乐一边开车。Or 他一边开车一边听音乐。

2a. I like sleeping while listening to music. (It emphasizes the situation in which the person is listening to music)

我喜欢听着音乐睡觉。

b. I like sleeping and listening to music at the same time (I like sleeping while listening to music, or I like listening to music while sleeping) - different meaning

我喜欢一边听音乐一边睡觉。Or 我喜欢一边睡觉一边听音乐。

Learn Mandarin Grammar: Verb +着 vs. 在+ Verb:

Verb  +  着: As the continuous state, it is used to explain how a subject has been left in a place in a certain position (continuously).

Examples:

1. I am reading a story book

a. 我看着故事书。(continuously)

b. 我在看故事书。(now/currently)

2. She is cooking potato pie

a. 她做着土豆饼。(continuously)

b. 她在做(zuò)土豆饼。(now/currently)

3. He is listening to classical music

a. 他听着古典音乐。(continuously)

b. 他在听古典音乐。(now/currently)

Learn Mandarin Grammar : 很+Adjective vs. Adjective + 着呢

很+Adjective vs. Adjective +  着呢:

Adjective +着(zhe)呢(ne) is used to indicate a high degree, with a tone of exaggeration (excessiveness).

Examples:

1. I am very busy.

a. 我很忙。

b. 我忙着呢。(more emphasize than 很)

2. This kind of dish is very delicious.

a. 这种菜很好吃。

b. 这种菜好吃着呢。(more emphasize than 很).

3. This clothes is very beautiful.

a. 这件衣服很漂亮。

b. 这件衣服漂亮着呢。(more emphasize than 很)