What are some things I can do to reduce the amount of trash my family and I produce?

Always recycle, would be the first thing to do. The next thing would be to not throw out cups you get from stores (Styrofoam, paper, or plastic) or restaurants, but to re-use them. And always try to make use for things you'd normally throw away.

Here's a two-for-one for you.

You can begin eating healthy....honestly fresh whole foods are the answer.
Fresh fruit, vegetables, bulk nuts, bulk seeds, etc.... do not come in containers.
Reduce if not eliminate all pre-packaged foods.

Look in your garbage can and see what items are there. Most will probably be food packaging. Drink cans, bottles, boxes, jars, bags, cartons, etc.

Transition to eating fresh whole foods and the packaging and consequentially the garbage disappears.

Stop paying for silly things like poorly flavored sugar water in a can, no matter how much advertising the company does it still doesn't taste good and its not good for you.
Plus it adds to the waste.

The more you transition to fresh whole foods the more trash disappears. It is amazingly simple with multiple benefits all around and the food tastes so much better.

After the transition my house puts out about one small grocery store plastic vegetable sack of trash a week.
That's a week.
Most of the trash now consists of mail advertising.

We're sleeping sounder, arguing less, feeling better, dropping weight and getting healthier (based on vital signs and appearance).
Plus we put out far less waste.

I Hope it helps.