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.
The only country in Europe you can visit without a visa is Serbia:
For holders of ordinary passports and other travel documents: No visas required for visits of up to 30 days, within period of one year
You can also obtain a 15-day visa on arrival if flying into Kyiv-Boryspil (KBP), Kyiv-Zhuliany (IEV), or Odessa (ODS) in Ukraine, subject to some document requirements.
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.
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.