I am not familiar with what bars are in the city, but the close proximity of both Nagoya and Gifu is probably responsible for the lack of bars.
If you've only explored Inuyama and Nagoya, then there are many places you can go. Of course, it depends on how far is "too far" for you, but even if you limit yourself to the Meitetsu Lines and nearby JR lines, there are still many options.
What are your interests?
Gifu City is very close and easy to access from Inuyama. It has some museums, a nice castle, and onsen, historic town and in the evening you can watch comorant fishing (although I think you can do this in Inuyama, too).
Ogaki has a nice castle and a museum dedicated to Matsuo Basho.
Sekigahara has a battle site.
Seki City is famous for its knives and has a museum and other sites related to knives.
Mino has a nice historic district and a castle.
Gujohachiman has many sites:
http://www.japan-guide.com/e/e5930.htmlGero is a good onsen option:
http://www.japan-guide.com/e/e5990.htmlTajimi has the scenic Eihoji Temple and a few other sites.
You can easily go to the Kiso Valley from Inuyama for hiking and/or to see the old districts along the Nakasendo Path:
http://www.japan-guide.com/e/e6075.htmlIn Aichi:
Komaki City has a castle and Menard Art Museum. It is also home to Tagata Shrine, one of Japan's famous fertility shrines:
http://members.virtualtourist.com/m/a40e1/edde5/Seto is one of Japan's most famous and respected pottery towns. There are sites there related to its pottery and you can try to make your own, I think.
Nagakute is home to the former Worlds Fair park.
Toyota has sites related to its car industry and the Toyota Art Museum is one of Aichi's top art museums.
You could explore the Chita Peninsula.
Visit Nishio for its castle and historic district or go to Sakushima, Aichi's art island.
There are lots of other places, too: Okazaki is historic, Toyokawa has a famous Inari shrine, etc.
You can easily go even further, too, to Hikone, Matsumoto, Takayama, and places in Shizuoka or Mie Prefectures.
If you are concerned about cost and your trip can be made using JR trains within the tickets area, the Aozora Free Kippu is a great option for weekend daytrips in the region:
http://www.japan-guide.com/e/e2357_004.htmlThere are also buses to places like Shirakawa-go that can save you a lot of money to visit. Most of them depart from Nagoya (some from Gifu).