How to use the phrase "'rather than"
I'd like to know which sentence is correct? and what are the exact rules of forming the verb right after "rather than"? ?

1. Thanks to more sports facilities, we could make exercise a part of our lives rather than stare at TV screen every day.
2. Thanks to more sports facilities, we could make exercise a part of our lives rather than staring at TV screen every day.

thank you.

Question

How to use the phrase "'rather than"
I'd like to know which sentence is correct? and what are the exact rules of forming the verb right after "rather than"? ?

1. Thanks to more sports facilities, we could make exercise a part of our lives rather than stare at TV screen every day. 
2. Thanks to more sports facilities, we could make exercise a part of our lives rather than staring at TV screen every day.

thank you.

Answer

I'm not an expert, but to me the verbs on either side of the 'rather than' should be the same. So - either 
we could make exercise a part of our lives rather than stare at the TV screen every day - or
we could be making exercise a part of our lives rather than staring at the TV screen every day. 
Note that your example is missing an article before 'TV':  it should be 'the TV', or possibly 'a TV'.