what are differences between "I'd prefer" and "I prefer"?
I also don't quite clear how to use "I'd prefer ... rather than..." to write a sentence. Which one of the following is correct?

I'd prefer playing soccer rather than watching TV at home.
I'd prefer to play soccer rather than to watch TV at home.
I'd prefer play soccer rather than watch TV at home.

Thank you for your answer in advance.

Question

what are differences between "I'd prefer" and "I prefer"? 
I also don't quite clear how to use "I'd prefer ... rather than..." to write a sentence.  Which one of the following is correct?

I'd prefer playing soccer rather than watching TV at home.  
I'd prefer to play soccer rather than to watch TV at home.  
I'd prefer play soccer rather than watch TV at home.

Thank you for your answer in advance.

Answer

"I prefer" is simple present tense.  When you say "I prefer" you are saying "I always like," because sentences in the simple present are habitual, or always true.

I would prefer (I'd prefer) is a conditional, and this means that your preference relies on something else.  I'd prefer is often used with "rather than" or "over" or "to"
Example:
I prefer coffee over tea
I prefer coffee to tea
I prefer coffee rather than tea

Since prefer ALWAYS takes a gerund (-ing verb), your first sentence is correct.

Answer

The 'd. And actually that's what you should ask about.