I think suggesting places to visit in Georgia is my favorite topic ever. Tbilisi is a great place to stay i Georgia. Places I would suggest to visit are old parts of the city which have beautiful architecture. Also botanical gardens are one of my favorite places to visit though I have never visited them in Autumn. You can also visit national museum there is planty to see.
Also seeing as you will be visiting Georgia in Octomber in time of rtveli which is time when grapes are collacted and wine is made, I would suggest visiting the best place to enjoy rtveli and try wine Kakheti. It is the whole experience really, plus if you will be able to experience Georgian sufra that will be a feast you never forget. You can use public transportation to get to kakheti but I would suggest renting a car in Tbilisi if you have drivers license.
One of the cultural experiences you must have in Georgia is trying national food like Khinkali, which are delicious dumpling meal, Lobiani and Khachapuri. Also some Georgian Wine and Chacha if you enjoy drinking alcohol and churchkhela if you like sweet treats.
If you like watching national dances I would suggest visiting sukhishvili concert in Tbilsi, national dances are very elegant and can very very fierce too, depending on a dance. You would also get the chance to listen to Georgian music.
And for the last if you would like to just go out and have some fun in bars I would suggest Wine Factory for drinks (there are many bars that serve cocktails and different drinks) and nearby is Brose if you want to dance a little to 2000s songs. And if you enjoy techno music and rave there are events at Khidi and Mtkvari and also an event called Mzesumzira.
I think I have overed all I know about, hope yo enjoy Georgia <3