Fort Worth has a wonderful cultural area and many museums have discounted entry fees. Check with your local museums to find out if they offer discounted or free tickets for certain days.
Modern Museum - Free on Sundays, half price on Wednesdays (children under 12 free) and free parking
Kimbell Art Museum - Visit to the permanent collection free. Half price tickets on Tuesdays and Fridays (check site for specific times) Parking is limited but you can park across the street at the Modern Museum.
Amon Carter Museum of American Art - Free admission but paid parking
National Cowgirl Museum - Discount for seniors and children (no free parking)
Texas Civil War Museum - (children 6 and under free) free parking
White Settlement Historical Museum - I believe the entry is free but donations are welcome. It is run by the White Settlement Historical Society. They often have free events such as re-enactments and musical evenings.
Lake Worth Foster House Museum - I believe this is free but donations accepted