In Louisville 

Locust Grove - http://www.locustgrove.org/index.htm
Locust Grove is a national historic landmark on 55 acres. In addition to many other events, it has a garden fair over Mother's Day Weekend and an 18th Century Market fair in the fall.

Farmington - http://www.historichomes.org/farmington/
Includes a working blacksmith shop.

Metro Parks - http://www.louisvilleky.gov/MetroParks/recreation/summerpools.htm
Look for pools and sprinklers here.

Further Away

Jeffersonville Aquatic Center - http://www.jeffparks.org/aquatic_center.html
This one is right across the river.  Lots of fun for hot, muggy Ohio valley days.

Falls of the Ohio - http://www.fallsoftheohio.org/
This one is also right across the river. They have several family festivals, including the Archaeology Day and the Fall Fossil Festival. Kids love to run along the fossil beds when the river is really low.

Bernheim Arboretum & Research Forest - http://www.bernheim.org/
A short drive from Louisville, this forest has multiple shorter trails that are suitable for young children. Admission is free during the week, but a nominal charge of $5 per passenger car is assessed on weekends and holidays.

Mammoth Cave National Park - http://www.nps.gov/maca/
In addition to being the world's longest cave system, this national park offers cave tours, surface hikes, canoeing on the Green River, picnicking, horseback riding, bicycling, camping and more. The campsites are clean and well managed.

Kentucky Down Under - http://www.kdu.com/
This is an Australian themed animal park with a natural onyx cave. There are kangaroos to pet, sheep dog herding demonstrations, as well as birds, reptiles, and cave tours.

Red River Gorge - http://www.fs.fed.us/r8/boone/districts/cumberland/redriver_gorge.shtml
The many activities include rockclimbing, backpacking, canoeing, hiking and camping.

Berea - http://www.bereacraftfestival.com
A town with Appalachian art, music, and food. They hold an annual craft festival in July.

Shaker Village - http://www.shakervillageky.org/
You can easily do a day trip or stay for a weekend. It's fun during the day and super peaceful to walk around after they close for the day. They have a restaurant with wonderful food. There is a family fun overnight in June and the Apple Harvest Weekend in September. They have a riverboat ride and there is a beautiful waterfall tucked back in the woods right beside the riverboat landing.

KY Horse Park - http://www.kyhorsepark.com/
Kids love horses.