You have to visit the botanical garden of Batumi. It opened in 1912 and you have never seen a garden as big as this one. Therefore they have an electrical train running through the park for 2 lari. Entrance fee for foreigners is 6 lari. To get there you need to go with marshrutka 31 (not 1 as lonely planet says). The marshrutka you can catch from Chavchavadzis street bus stop or Gogebashvilles. It is 9 km north of Batumi. Afterthe visit you can swim in the sea there.
In the garden you have spectacular views of Batumi and the sea.
You should really visit the Gonio fortress. It is a very interesting place with a nice small museum. Take a marshrutka from Batumi to Sharpi and pay 1 Lari. Or go with bus number 101 buy a bus ticket two rides for 0,80. Ask the people to help you get out at Asparos. They only understand this word. It is on the left side at the beginning of the village. In the castle you can find the grave of the apostle Matthias. The gold they found here is displayed at the archeological museum of Batumi. All the buses leave from the TBilisi moedani.
After your visit you can have a good swim in the sea overthere.
While waiting for the sun to come out, I took a walk along Batumi's promenade to the south. After a while, the park runs out and is replaced with huge tower blocks, quietly rusting away. A couple of the buildings used to be hotels, but are now filled with refugees from Abkhazia and look squalid and depressing. The promenade is being extended, and a park being constructed, which will hopefully make the area look a little less bleak...but for now you can admire some of the colourful grafitti in between the grotesque blocks of flats.