I plan to go to Kars in July and would like to know the best way to travel from there to Tblisi by train or bus. 

I did it many years ago so I don't remember precisely anymore, but I think I took a bus from Kars to Ardahan, and then hitchhiked from Ardahan to Posof and onward to Akhaltsikhe. At the time there was no bus connection between Kars and Georgia, and I doubt it that it has changed.

A longer, but probably easier way would be to take a bus from Kars to Hopa (Black Sea coast) and hitchhike or get on a bus to Batumi. The coastal road is much busier, so even though it looks farther it might actually be quicker. And there should be buses, coming all the way from Istanbul and Trabzon. 

Hope that helps.