Someone asked me: "[What do you] do on the sabbath day that make it meaningful to you?"
It is my present understanding that we ought to be constantly occupied in whatever leads to the greatest benefit, all the time. What that is, exactly, is obviously specific to a person, due to variance in capability and understanding.
Sundays are no different in purpose, but for most people, the things they do that day may be different in practice because of relaxation of day-to-day requirements.
The sabbath is a gift from God because we get to focus for more time on the things that the necessity of earning one's daily bread forces us to limit during the week.
Each day, from when I get up until when I go to sleep, I prioritize my actions to do the greatest good I know of, subject to the limits of my mortal body.
Every day I wake up as early as I can. I spend the time until 6:30 writing, studying, praying, and lifting weights inasmuch as it helps me focus my mind.
At 6:30, I spend thirty minutes teaching my children out of the scriptures.
At 7, I shower and go to work.
Throughout the day, whenever God speaks to me, I stop whatever it is I am doing and write down what he tells me until he has nothing left to tell me. I have notepads everywhere where I don't have access to a computer.
When I get home, I eat dinner, then give my family my attention for the rest of the night, unless something is broken and needs to be fixed and can't wait until Saturday.
The weekends are slightly different.
On Sunday, I don't go to work. Instead, I write until I am so exhausted that I can't write another word. How long this takes varies. Sometimes I can go until dinnertime. Most days I run out of gas around 2pm. At that point I make a point to spend some time with my family. This schedule is overriden whenever God tells me to do something different.
Before I had so much to write about, I would spend some of the time I now occupy in writing by visiting or calling family members, especially older or lonely people, doing service for others, and reading the scriptures. All of my grandparents are now dead, there is no greater service I can do than the writing I am presently doing, and I still spend a lot of time in the scriptures.