I'm an Indian citizen and do have an Indian passport, presently studying in Georgia, and I possess a temporary resident visa of Georgia for a duration of 5 years which was allotted for study purposes. Yearly we will be getting vacations. So I was thinking about visiting possible countries which I can visit without the requirement of a visa. So if someone could help me out with the list of countries which an Indian passport holder with temporary residence of Georgia can visit, it would be very helpful.
Georgia is not a member of the EU or EEA, so residence there does not really help you as far as visa exemptions are concerned.
You can also obtain an e-Visa for the neighboring countries of Armenia and Azerbaijan.
Otherwise, from checking Timatic and Wikipedia, you will need to obtain a visa in advance for anywhere else in Europe.
As of 2023, the only European countries that are visa-free for holders of Indian passports are Albania and Moldova. Both are quite "off the beaten track", so it'll require some research to get there without crossing any other countries which require visas from Indians (some Schengen countries - not all of them! - even require Airport Transit Visas from Indian passport holders).
If we use a wider definition of Europe, then you can also visit Northern Cyprus visa-free. Additionally, Armenia and Azerbaijan can be visited with an eVisa.
Your Georgian visa doesn't help you at all. A Schengen visa or a United States visa (or residence permits) would help you in some European countries.
According to Timatic(this version on the Emirates web site) you'll need a visa to enter Britain, Ireland and the Schengen area countries. That's the major EU countries - I haven't checked the rest, but it's likely to be similar. If you have a specific destination in mind, please edit into your question.
[Edit] I've checked the remaining EU countries and Switzerland - none will allow you to visit without a visa. Your Georgian residency doesn't help you.
You might find this list of countries that offer visa-free or visa-on-arrival travel for Indian citizens.
If you're thinking about a non-european destination in mind you can check the visa requirements yourself on the Timatic page I linked above.