I like doing - 하는게 좋다, 하는걸 좋아한다

I like doing - 하는게 좋다

하는게 좋다 is a contracted form of 하는 것이 좋다 (것이 → 게)
하는 것 means "doing" and if you remember, "가/이" is an identifier particle. I'd also like to call it a specificity particle.

For example,

• 점심에는 샌드위치 먹는 것이 좋다 = I like eating a sandwich at lunch (among a variety of things to eat at lunch, a sandwich is my choice)
  
• 시골에서 사는게 좋다 = I like living in the country (Although I could live in the city or urban area, I like living in the country)

좋다 means two things

• 1. I like
• 2. It's good

Usually, its meaning is "I like doing" in spoken Korean. In written Korean, "하는게 좋다" is more likely to mean "it's good to do something." However, the meaning depends on the context.

Informal	Positive	Negative
Written	하는게 좋다	하지 않는게 좋다 안하는게 좋다
Spoken	하는게 좋아	안하는게 좋아

Formal	Positive	Negative
Written	하는게 좋습니다	하지 않는게 좋습니다 안하는게 좋습니다
Spoken	하는게 좋아요	하지 않는게 좋아요 안하는게 좋아요

Note: The words in bold are the more commonly used alternatives of the two in each box.

Rules
1. Take 다 off a plain form of verbs and then attach 는게 좋다
2. If a verb has a final consonant of ㄹ, it is omitted.

• 하다 → 하는게 좋다 = I like doing, it's good to do
  
• 먹다 → 먹는게 좋다 = I like eating, it's good to eat
  
• 보다 → 보는게 좋다 = I like watching, it's good to watch
  
• 듣다 → 듣는게 좋다 = I like listening, it's good to listen
  
• 걷다 → 걷는게 좋다 = I like walking, it's good to walk
• 쓰다 → 쓰는게 좋다 = I like writing, it's good to write
  
• 읽다 → 읽는게 좋다 = I like reading, it's good to read
• 만들다 → 만드는게 좋다 = I like making, it's good to make
• 살다 → 사는게 좋다 = I like living, it's good to live
• 말다 → 마는게 좋다 = I like wrapping, it's good to wrap

Example sentences 

• 운동하는게 좋다 = I like exercising. It's good to exercise.
  
• 휴일에는 바닷가에 가는게 좋다 = I like going to the beach at weekends. It's good to go to the beach at weekends.
• 금요일밤에는 피자 먹는게 좋다 = On a Friday night, I like eating pizza. On a Friday night, it's good to eat pizza.
  
• 심심할땐 영화보는게 좋아 = When I'm bored, I like watching movies. When you're bored, it's good to watch movies.
  

I like doing - 하는걸 좋아한다

하는걸 is a contracted form of 하는것을 and 을 is an object particle.
하는걸 좋아한다 just means "I like doing"

Informal	Positive	Negative
Written	하는걸 좋아한다	하는걸 좋아하지 않는다 하는걸 안좋아한다
Spoken	하는걸 좋아해	하는걸 좋아하지 않아 하는걸 안좋아해

Formal	Positive	Negative
Written	하는것을 좋아합니다	하는것을 좋아하지 않습니다 하는걸 안좋아합니다
Spoken	하는걸 좋아해요	하는걸 좋아하지 않아요 하는걸 안좋아해요

Note: 걸 is a contracted form of 것을.

Rules are the same as above.

• 하다 → 하는걸 좋아한다 = I like doing
• 먹다 → 먹는걸 좋아한다 = I like eating
• 보다 → 보는걸 좋아한다 = I like watching
• 듣다 → 듣는걸 좋아한다 = I like listening
• 걷다 → 걷는걸 좋아한다 = I like walking
• 쓰다 → 쓰는걸 좋아한다 = I like writing
• 읽다 → 읽는걸 좋아한다 = I like reading
• 만들다 → 만드는걸 좋아한다 = I like making
• 살다 → 사는게 좋아한다 = I like living
• 말다 → 마는게 좋아한다 = I like wrapping

Example sentences

• 운동하는것을 좋아하지 않는다 = I don't like exercising.
• 휴일에는 바닷가에 가는걸 좋아한다 = I like going to the beach at weekends.
• 금요일밤에는 피자 먹는걸 좋아합니다 = On a Friday night, I like eating pizza.
• 심심할땐 영화보는걸 좋아해요 = When I'm bored, I like watching movies.