Compared to other Indian cities Bangalore has a dearth of cultural / historic sites that would be of huge interest to tourists. There are a couple of good parks, and Lalbagh in particular is quite good for a leisurely stroll looking at the various botanical exhibits they have. I'm honestly not sure a visit to the Bangalore palace is worth it for someone with just one day in town, unless you are really quite interested in the history of the place. It's not a very historic place - it was built in the late 1800s in a Tudor style, so nothing traditional or deeply culturally connected. A visit to the Bangalore Museum may be better if seriously interested in local history, although be warned that it is a very small museum and you'll be done looking at all the exhibits within a couple of hours at most. A trip to Mysore would give you much more of the region's history (Mysore was the seat of the kingdom that ruled the area) but it will take up most of the day.
If you don't feel like a day trip out of town, other options are to walk around some of the lively areas of the city. The area around M G Road, Church Street, Brigade Road is quite busy with lots of places to eat, some funky bookstores, etc., but with a more "modern" feel. You could go further to wander around the K R Market area or Gandhi Bazaar to get a feel for the more traditional parts of the city, with their bustling areas of commerce for groceries and clothes and such.
The zoo or nature preserve (Bannerghatta) can be interesting - they have a fair number of tigers and such there in a somewhat open environment. But it's hit and miss how many of them you will see on the "safari" trip where they take you in a covered bus through the park (need to buy tickets and show up early for this), and some of the other animal exhibits in the park are not very well maintained, which might actually be somewhat upsetting if you are an animal lover.