Doing the "right thing." Not being a racist. Not putting others down for their weaknesses, instead raising them up. Treating those who have less money, less strength, less social standing (whatever that means to you), less power, less prestige, less of a voice, less opportunity, less whatever as an equal to you (which they are). Treating others with respect, helping others when you can. Caring for animals, the elderly, the poor, children, the sick, anyone who needs caring- within your means to do so. Not taking advantage of or hurting anyone- mentally, physically, or emotionally. Allowing people and animals to pursue happiness and their own free will. Helping those in need if they need food, water, or shelter- animals or humans. Taking care of our environment, doing your part as you are able. That is morality to me.

Raised a Baptist, made to memorize bible verses and go to church Sundays, but this is pretty much what has defined me, since I first read it as a girl:
Abou Ben Adhem, by Leigh Hunt

Abou Ben Adhem (may his tribe increase!)
Awoke one night from a deep dream of peace,
And saw, within the moonlight in his room,
Making it rich, and like a lily in bloom,
An angel writing in a book of gold:—
Exceeding peace had made Ben Adhem bold,
And to the presence in the room he said,
"What writest thou?"—The vision raised its head,
And with a look made of all sweet accord,
Answered, "The names of those who love the Lord."
"And is mine one?" said Abou. "Nay, not so,"
Replied the angel. Abou spoke more low,
But cheerly still; and said, "I pray thee, then,
Write me as one that loves his fellow men."

The angel wrote, and vanished. The next night
It came again with a great wakening light,
And showed the names whom love of God had blest,
And lo! Ben Adhem's name led all the rest.

I agree with your grandmother.... morality is making a conscious choice to do the right thing in a given situation - even when it is not the easy or expedient or self serving choice. And since we are all human, after all, some situations are more difficult to deal with than others so the right choice may well be the more difficult one.

Morality is how society decides what sustainably makes it the way most of its members want it to be -- generally, that means civil, humane, respectful, safe and stable. The first time two or more people needed to coexist or cooperate, morality was born. Before that, it had no meaning.

I would say it has more to do with harms or benefits than whether something is true, although obviously it's hard to have anything but harms in the presence of untruth.

As best as i can describe it.

There is nothing wrong with learning morality from a religious relative..most of us were raised by devoutly religious relatives.
They would have passed on a lot of wise and important gems of perfectly good rational and altruistic moral codes and standards . For example a common Christian moral theme which has crossed over to Humanism is the idea of treating people the way you would like or expect to be treated..there are many more of course from all religions.
You may reject the dogma..but keep the human moral sentiments with you...