Hey Anwesha, I would suggest you to visit hill stations like Coorg or Wayanad or Ooty and even Kodaikanal ... Your pet would have a lot to roam around and even you will get a splendid time to walk around nature ... Happy Travelling!!!
Well, if it is a day trip around your own city (which I mean you have your own car) and your pet is trained to travel with you then it seems quite comfortable and fun with the four legged friend. But make sure that you will have to take multiple breaks for your pets' pee and poo break as they may not be able to hold too long. I don't think many places like malls or museums allow pet friendly entry and keeping your pet in the car is a big NO due to safety concerns. Maybe you can try taking your pet to pet friendly cafes which is quite in trend around Bangalore. Therpup is one such cafe for where dogs are allowed!