It is a park at the Mtatsminda Mount. A perfect place: fresh and clean air, astonishing view on the city. The park is with its history and various attractions; the only downside is the awful fast food.
There used to be amusement park here, and now it is just a place for walking with several nice bars and restaurants (including club of Niaz Diasamidze “33a”). In the mornings you can see there lots of lovely dogs!
This place is under grand reconstruction now: there will be swimming pool, palace of weddings, chess club, parkour, skating and carting equipment. Later it all will be connected by bridge with former zoo and will become the analogue of Central Park in New York.
My favorite place for morning jogging. It is apparently in the center of the city, but wandering serpentine takes you high above the sea level. Here you can find everything: fresh air, amazing views, lake, dogs, bicycle riders, colorful people.
Tbilisi Botanical Garden
There were times when old park went through lack of financing and attention, but nowadays it is quite well maintained. The place is amazing, very high, and the area is colorful: the city view, flowers, relax zones. There is also a waterfall and you can even swim there.