In mountain resort Kojori, which is 17km far from Tbilisi, there is a historical meaning hotel "The Vorontsovs" , which used to be the residence of governor of Georgia in 1850th.
The place is very beautiful, in Kojori there is a big forest, very nice cool climate and very good places for hicking, hunting, etc.
Hotel costs around 40 - 45 Euros, with very good continantal breakfast, if you like you can take all incusive (breakfast, dinner, supper) room price will be 65 Euros.