A fFamily time

As a busy student, I barely know what ist means to enjoy a family time.

My life's always been hectic and rushed, and I rarely get a chance to have a meal with all family members together.

You can also say: With all of my family together or with all family members present

Sometimes, I've felt like I've become distant from them and the dinner table was always cold and quiet - awkward moments that were hard to avoid.

But on this particular occasion, I learned to understandappreciate the joy and warmth of having a family moment together.

You can also say: understand the meaning of the joy...

Jokes and laughter filled the enthusiastic atmosphere.

Go with the tense you have already chosen when listing things. Example: all members are present, jokes and laughter fill or all members were present, jokes and laughter filled

A fFamily tTime

In this context, "time" is uncountable. Alternatively: "A Time Spent with Family". In this case, "time" is countable, because it refers to a specific period of time (in the past).

As a busy student, I barely know what is mean tot by enjoy aing family time.

(1) Alternatively: "I barely know what it means to enjoy family time."
(2) Alternatively: "I barely know what is meant by enjoying time with family."

My life's always been hectic and rushed, and I rarely get a chance to have a meal with all family members together.

rushed: done or completed too hurriedly; hasty. Here, "rushed" is an adjective, not a verb.

Sometimes, I felt like I'd become distant from them and the dinner table was always cold and quiet - awkward moments that were hard to avoid.

The past perfect ("I'd") is expected here, because there are two past events we're talking about—you becoming distant, THEN you feeling like you'd become distant.

All family members awere present at the dinner table.

You're describing a past memory, so the past tense is expected.

Grandparents shared their reminiscences of their lives.

(1) Again, mind the tense.
(2) "Reminiscence" should be plural, since multiple grandparents implies multiple reminiscences.

All awere having their meals happily while chatting.

Alternatively, and perhaps more naturally: "Everyone was having their meals happily while chatting."

I listened with great fascination, eager to learn (more) about them.

"Learn about them" is okay, but you can consider "learn more about them", which I find, in my own experience, is a more common expression. "Learn about them" might suggest that they were complete strangers.

It was the most memorable moment in my life and I want to cherish this momenit forever.

You can consider replacing "this moment" with "it" to avoid repetition of the word.

Perhaps, there will be more of this kind of family times in the future.

The comma breaks the flow of the sentence is an awkward way.

A fFamily time

As a busy student, I barely know what ist means to enjoy a family time.

My life's always been hectic and in a rush, and I rarely get a chance to have a meal with all family members together.

Sometimes, I feelt like I have become distant from them and the dinner table was always cold and quiet - awkward moments that were hard to avoid.

But on this particular occasion, I learned to understand the joy and warmth of having a family moment.

Jokes and laughter filled the enthusiastic/ lively atmosphere.

Grandparents share their rmeminiscenceories of their lives.

All are having their meals happily while chatting.

"All are..." makes senses, but I would rather say "Everyone is... "

I listened with great fascination, eager to learn more about them.

It was the most memorable moment in my life and I want to cherish this momenit forever.

Perhaps, there will be more of this kind of family times in the future.