Day trips to the region of Kakheti in Georgia are quite possible and I would actually recommend taking one if you have a chance. I have explored the region of Kakheti in different ways (mostly by marshrutka and car). I prefer a personal car with a driver. If you feel you are indie or rough traveler you might try marshrukta leaving from Tbilisi.
Today I'll show you one of my favorite Kakheti day trip routes which is accessible by car.
This route will take you from Tbilisi to Telavi, then to nearby Kvareli and in the evening back to Tbilisi. (You should plan this day trip as early as possible)
It might seem like an easy route, but it will take all your day (especially if you will take your time to take pictures at these sites). You will visit a few really interesting Georgian Orthodox Churches (Nekresi monastery complex is among my favorite), a wine tunnel (you can have a wine tasting here, for additional money), you will see a Giant Plane Tree (more than 900 years old and about 12 meters wide/ 46 meters high). I would suggest trying local Telavi khinkali at Zodiako bar, and if you will be lucky enough (the day won't be over yet) you could make a stop at David Gareja Monastery complex.
I was able to manage such day trip for about GEL 80 + gas and food expenses for my driver.
Enjoy your travels in Georgia!