Jardin du Luxembourg

The Jardin du Luxembourg has everything a public garden should have. There is a large lawn shaded on both sides by thick rows of groomed chestnut trees. Down some grand stairs is a huge circular pool where toy boats are sailed. In front is a palace. There are sculptures everywhere, among the trees and flowers. Flower beds of the loveliest kind are all around the park- check out the tulips in early May. There are tree canopies that provide walkways with thick shade.

For kids there's a wonderful puppet theater (marionnettes) creating precious memories (see times below)! There's also a fantastic playground called Poussin Vert. You pay to enter (see price below) and then get to enjoy every possible kind of play equipment you could imagine, all shaded by tidy trees.
In other spots in the park there are also swings, a carousel, pony rides, and snack stands.
Easy to miss is the Medici fountain, pretty with its reflecting pool and surrounding foliage. Hidden by trees, it's in the northeast corner of the park.
The Sorbonne is nearby so there are many students who hang out in the park, giving it a hip vibe.
The garden was created in 1612 and is owned by the French Senate, which meets in its palace.
One thing that's not so nice about the park is that the walkways are loose white dirt that gets kicked up by the joggers that go by, making a cloud of dirt you then breathe.
After exploring the garden, walk up groovy Boulevard Saint Michel and enjoy a large (this is rare in France!) coffee or tea at Costa Coffee, or a sandwich at one of the many cafes.
Other amazing parks in Paris: Parc des Buttes Chaumont, and Jardin des Tuileries, and Parc Montsouris.
If you love gardens, you will love the statues, topiaries, orangerie, reflecting pools, fountains and secret groves at Versailles Gardens. 

Directions

The Jardin du Luxembourg is located in the 6th arrondisement. Luxembourg is the nearest metro station.
The puppet shows are at 3pm on Wed, 11am and 3pm on Sat and Sun. Pay 6 euros each at the booth in front of the theater. During the holidays there are extra shows every day at 3pm and sometimes 2pm as well. Check the times and show titles here.
You can rent toy boats from a little kiosk by the pond.
The playground is open from around 10am-7pm (varies with the season). Entry costs 2.50 euros for children aged 15 months to 16 years, and 1.20 euros for adults. Children under 15 months old enter for free. Large families can get a discount card. Disabled children enter for free. No smoking is allowed at the playground.

Click on map for interactive view